Universally optimal (UO) codes were introduced by H. Cohn and A. Kumar in 2007 and later extended to the discrete setting by Cohn and Y. Zhao. They minimize the ``energy'' among all codes of the same size for a certain class of potential functions. So far only a small number of specific functionals have been linked to information theory problems. We add one more, showing that UO codes optimize $α$-mutual information for $α=2$ in the context of wiretap channels with a noiseless main channel. This implies that UO codes are optimal for transmission over this type of wiretap channels.

Designing efficient and noise-tolerant quantum computation protocols generally begins with an understanding of quantum error-correcting codes and their native logical operations. The simplest class of native operations are transversal gates, which are naturally fault-tolerant. In this paper, we aim to characterize the transversal gates of quantum Reed–Muller (RM) codes by exploiting the well-studied properties of their classical counterparts. We start our work by establishing a new geometric characterization of quantum RM codes via the Boolean hypercube and its associated subcube complex. More specifically, a set of stabilizer generators for a quantum RM code can be described via transversalXandZoperators acting on subcubes of particular dimensions. This characterization leads us to definesubcube operatorscomposed of single-qubit$\pi /2^{k}~Z$-rotations that act on subcubes of given dimensions. We first characterize the action of subcube operators on the code space: depending on the dimension of the subcube, these operators either (1) act as a logical identity on the code space, (2) implement non-trivial logic, or (3) rotate a state away from the code space. Second, and more remarkably, we uncover that the logic implemented by these operators corresponds to circuits of multi-controlled-Zgates that have an explicit and simple combinatorial description. Overall, this suite of results yields a comprehensive understanding of a class of natural transversal operators for quantum RM codes.

We improve the secrecy guarantees for transmission over BMS wiretap channels that relies on regular LDPC codes. Previous works (Thangaraj e.a., 2007, 2010) showed that LDPC codes achieve secrecy capacity of some classes of wiretap channels while leaking$o(n)$bits of information over$n$uses of the channel. We improve the security component of these results by reducing the leakage parameter to$O\left(\log ^{2} n\right)$. While this result stops short of proving strong security, it goes beyond the general uniformity properties of capacity-approaching code families.

We study recoverable systems on the 2D lattice using tools from statistical mechanics. For a particular recovery rule that we consider as an example, we compute lower and upper bounds on the topological entropy of the system (the case of zero temperature). We also show that at low positive temperature, typical configurations are local perturbations of the ground states. For the case of high temperature, we show uniqueness of the Gibbs measure and find the mixing rate of Glauber dynamics for the system in a finite domain.

We consider the problem of distilling uniform random bits from an unknown source with a givenp-entropy using linear hashing. As our main result, we estimate the expectedp-divergence from the uniform distribution over the ensemble of random linear codes for all integerp≥ 2. The proof relies on analyzing how additive noise, determined by a random element of the code from the ensemble, acts on the source distribution. This action leads to the transformation of the source distribution into an approximately uniform one, a process commonly referred to as distribution smoothing. We also show that hashing with Reed-Muller matrices reaches intrinsic randomness of memoryless Bernoulli sources in thelpsense for all integerp≥ 2.

Universal hash functions map the output of a source to random strings over a finite alphabet, aiming to approximate the uniform distribution on the set of strings. A classic result on these functions, called the Leftover Hash Lemma, gives an estimate of the distance from uniformity based on the assumptions about the min-entropy of the source. We prove several results concerning extensions of this lemma to a class of functions that arek*-universal, i.e.,l-universal for all 2 ≤l≤k. As a common distinctive feature, our results provide estimates of closeness to uniformity in terms of the α-Rényi divergence for all α ∈ (1, ∞]. For 1 ≤ α ≤kwe show that it is possible to convert all the randomness of the source measured in α-Rényi entropy into approximately uniform bits with nearly the same amount of randomness. For large enoughkwe show that it is possible to distill random bits that are nearly uniform, as measured by min-entropy. We also extend these results to hashing with side information.

We consider regenerating codes in distributed storage systems where connections between the nodes are constrained by a graph. In this problem, the failed node downloads the information stored at a subset of vertices of the graph for the purpose of recovering the lost data. Compared to the standard setting, regenerating codes on graphs address two additional features. The repair information is moved across the network, and the cost of node repair is determined by the graphical distance from the helper nodes to the failed node. Accordingly, the helpers far away from the failed node may be expected to contribute less data for repair than the nodes in the neighborhood of that node. We analyze regenerating codes with nonuniform download for repair on graphs. Moreover, in the process of repair, the information moved from the helpers to the failed node may be combined through intermediate processing, reducing the repair bandwidth. We derive lower bounds for communication complexity of node repair on graphs, including repair schemes with nonuniform download and intermediate processing, and construct codes that attain these bounds. Additionally, some of the nodes may act as adversaries, introducing errors into the data moved in the network. For repair on graphs in the presence of adversarial nodes, we construct codes that support node repair and error correction in systematic nodes.

We construct a new family of permutationally in-variant codes that correct$t$Pauli errors for any$t\geqslant 1$. We also show that codes in the new family correct quantum deletion errors as well as spontaneous decay errors. Our construction contains some of the previously known permutationally invariant quantum codes as particular cases. In many cases, the codes in the new family are shorter than the best previously known explicit permutationally invariant codes for Pauli errors and deletions. This is an extended abstract of the preprint [1].

We study regenerating codes in heterogeneous distributed storage systems including the node repair problem in graphically constrained architectures. We show that the communication cost of repair can be decreased by downloading the amounts of data controlled by the distance of the helper to the failed node. At the same time, given the flexible choice of the repair degree, the optimal repair cost can always be attained by relying on uniform downloads. We also give a construction of codes that attain a general version of the cutset bound for heterogeneous and graphically constrained systems. The codes we construct also support data combining at intermediate nodes during repair.

Node repair on graphs is a recent variation of the distributed storage model, where connections between the storage nodes are described by a graph. Here we study this problem under the assumption that some of the nodes act as adversaries, altering the data that they store. We derive bounds on the communication complexity of repair and construct codes that support repair in the presence of adversarial nodes.

A storage code is an assignment of symbols to the vertices of a connected graph G(V, E) with the property that the value of each vertex is a function of the values of its neighbors, or more generally, of a certain neighborhood of the vertex in G. Under the name of recoverable systems, a class of storage codes on ${\mathbb{Z}}$ was recently studied relying on methods from constrained systems and ergodic theory. In this work, we address the question of the maximum capacity of recoverable systems on ${\mathbb{Z}}$ and ${{\mathbb{Z}}^2}$ from a combinatorial perspective. We establish a closed form formula for the capacity of several one- and two-dimensional systems, depending on their recovery set, using connections between storage codes, graphs, anticodes, and difference-avoiding sets.

We consider the use of regenerating codes in distributed storage systems where connections between the nodes are constrained by a graph. In this setting the cost of node repair is determined by the graphical distance from the helper nodes to the failed node. In our recent work (arXiv:2108:00939) we considered the MSR case, showing that linear MSR codes are amenable to intermediate processing of the information, resulting in reduced repair bandwidth which also meets the lower bound on the minimum repair cost. Here we extend this study to the non-MSR case. We derive a lower bound on the repair bandwidth and formulate repair procedures with intermediate processing for several families of regenerating codes, with an emphasis on the recent constructions from multilinear algebra. We also consider intermediate processing for the problem of partial node repair.

Consider an assignment of bits to the vertices of a connected graph$G(V,E)$with the property that the value of each vertex is a function of the values of its neighbors. A collection of such assignments is called a storage code of length$|V|$on$G$. The storage code problem can be equivalently formulated as maximizing the probability of success in a guessing game on graphs, or constructing index codes of small rate. If$G$contains many cliques, it is easy to construct codes of rate close to 1, so a natural problem is to construct high-rate codes on triangle-free graphs, where constructing codes of rate$> 1/2$is a nontrivial task, with few known results. In this work we construct infinite families of linear storage codes with high rate relying on coset graphs of binary linear codes. We also derive necessary conditions for such codes to have high rate, and even rate potentially close to one. We also address correction of multiple erasures in the codeword, deriving recovery guarantees based on expansion properties of the graph. Finally, we point out connections between linear storage codes and quantum CSS codes, a link to bootstrap percolation and contagion spread in graphs, and formulate a number of open problems.

Motivated by the established notion of storage codes, we consider sets of infinite sequences over a finite alphabet such that every$k$-tuple of consecutive entries is uniquely recoverable from its$l$-neighborhood in the sequence. We address the problem of finding the maximum growth rate of the set, which we term capacity, as well as constructions of explicit families that approach the optimal rate. The techniques that we employ rely on the connection of this problem with constrained systems. In the second part of the paper we consider a modification of the problem wherein the entries in the sequence are viewed as random variables over a finite alphabet that follow some joint distribution, and the recovery condition requires that the Shannon entropy of the$k$-tuple conditioned on its$l$-neighborhood be bounded above by some$\epsilon >0$. We study properties of measures on infinite sequences that maximize the metric entropy under the recoverability condition. Drawing on tools from ergodic theory, we prove some properties of entropy-maximizing measures. We also suggest a procedure of constructing an$\epsilon $-recoverable measure from a corresponding deterministic system.

We study the problem of erasure correction (node repair) for regenerating codes defined on graphs wherein the cost of transmitting the information to the failed node depends on the graphical distance from this node to the helper vertices of the graph. The information passed to the failed node from the helpers traverses several vertices of the graph, and savings in communication complexity can be attained if the intermediate vertices process the information rather than simply relaying it toward the failed node. We derive simple information-theoretic bounds on the amount of information communicated between the nodes in the course of the repair. Next we show that Minimum Storage Regenerating (MSR) codes can be modified to perform the intermediate processing, thereby attaining the lower bound on the information exchange on the graph. We also consider node repair when the underlying graph is random, deriving conditions on the parameters that support recovery of the failed node with communication complexity smaller than required by the simple relaying.

Motivated by the established notion of storage codes, we consider sets of infinite sequences over a finite alphabet such that every$k$-tuple of consecutive entries is uniquely recoverable from its$l$-neighborhood in the sequence. In the first part of the paper we address the problem of finding the maximum growth rate of the set as well as constructions of explicit families (based on constrained coding) that approach the optimal rate. In the second part we consider a modification of the problem wherein the entries in the sequence are viewed as random variables over a finite alphabet, and the recovery condition requires that the Shannon entropy of the$k$-tuple conditioned on its$l$-neighborhood be bounded above by some$\epsilon > 0$. We study properties of measures on infinite sequences that maximize the metric entropy under the recoverability condition. Drawing on tools from ergodic theory, we prove some properties of entropy-maximizing measures. We also suggest a procedure of constructing an$\epsilon$-recoverable measure from a corresponding deterministic system, and prove that for small$\epsilon$the constructed measure is a maximizer of the metric entropy.

We estimate the communication complexity of node repair for regenerating codes defined on graphs. Both deterministic and random graphs are considered.

There are few mathematicians whose contributions go beyond named conjectures and theorems: Vladimir Iosifovich Levenshtein (, 1935–2017) is one such true exception. During the five decades of his active research career, he enriched combinatorics, coding, and information theory with elegant problem formulations, ingenious algorithmic solutions, and highly original proof techniques. However, his work accomplished much more—it paved the way for the creation and advancement of new scientific disciplines, such as natural language processing, metagenomics, sequence alignment, and reference-based genome assembly, as well as DNA-based data storage, to name a few. A crucial concept behind sequence alignment algorithms used in phylogeny, comparative, and cancer genomics, as well as in natural language processing is the Levenshtein (edit) distance and its extension, termed the Damerau–Levenshtein distance between strings. The Levenshtein distance equals the smallest number of insertions, deletions, or substitutions required to convert one string into another. Levenshtein introduced this metric in 1965 [item 1) in the Appendix], followed by the notion of deletion and insertion error-correcting codes that have since been used in a myriad of systems presented with synchronization errors [items 1) and 2) in the Appendix]. Levenshtein’s work also inspired the introduction of the trace reconstruction problem [items 3) and 4) in the Appendix] which has since sparked substantial interest in the field of DNA-based data storage.

Locally recoverable (LRC) codes and their variants have been extensively studied in recent years. In this paper we focus on cyclic constructions of LRC codes and derive conditions on the zeros of the code that support the property of hierarchical locality. As a result, we obtain a general family of hierarchical LRC codes for a new range of code parameters. We also observe that our approach enables one to represent an LRC code in quasicyclic form, and use this representation to construct tail-biting convolutional LRC codes with locality. Among other results, we extend the general approach to cyclic codes with locality to multidimensional cyclic codes, yielding new families of LRC codes with availability, and construct a family of$q$-ary cyclic hierarchical LRC codes of unbounded length.

We address two aspects of the repair problem of Reed-Solomon codes. First, we propose a new repair scheme for the RS codes constructed in [Tamo-Ye-Barg, IEEE Trans. Inf. Theory, May 2019] which in addition to optimal repair bandwidth is also robust to erroneous information provided by the helper nodes. Next, we construct a new family of RS codes with optimal access for the repair of any single failed node. We also prove that any scalar MDS code with optimal repair bandwidth can be furnished with a repair scheme with the optimal access property.

Locally recoverable (LRC) codes and their variants have been extensively studied in recent years. In this paper we focus on cyclic LRC codes, presenting two results regarding codes with hierarchical locality and codes with availability. Regarding hierarchical LRC codes, we observe that the cyclic codes of Tamo et al. (2016) can be generalized to yield optimal families with multiple levels of locality for a broader range of parameters than known previously. We also observe that the general approach to cyclic codes with locality can be extended to multidimensional cyclic codes, yielding new families of LRC codes with availability.

The paper is devoted to the problem of erasure coding in distributed storage. We consider a model of storage that assumes that nodes are organized into equally sized groups, called racks, that within each group the nodes can communicate freely without taxing the system bandwidth, and that the only information transmission that counts is the one between the racks. This assumption implies that the nodes within each of the racks can collaborate before providing information to the failed node. The main emphasis of the paper is on code construction for this storage model. We present an explicit family of maximum distance separable (MDS) array codes that support recovery of a single failed node from any number of helper racks using the minimum possible amount of inter-rack communication(such codes are said to provide optimal repair). The codes are constructed over finite fields of size comparable to the code length. We also derive a bound on the number of symbols accessed at helper nodes for the purposes of repair, and construct a code family that approaches this bound, while still maintaining the optimal repair property. Finally, we present a construction of scalar Reed-Solomon codes that support optimal repair for the rack-oriented storage model.

Recently Reed-Solomon (RS) codes were shown to possess a repair scheme that supports repair of failed nodes with optimal repair bandwidth. In this paper, we extend this result in two directions. First, we propose a new repair scheme for the RS codes constructed in [Tamo-Ye-Barg, IEEE Transactions on Information Theory, vol. 65, May 2019] and show that repair is robust to erroneous information provided by the helper nodes while maintaining the optimal repair bandwidth. Second, we construct a new family of RS codes with optimal access for the repair of any single failed node. We also show that the constructed codes can accommodate both features, supporting optimal-access repair with optimal error-correction capability. Going beyond RS codes, we also prove that any scalar MDS code with repair bandwidth attaining the cutset bound affords a repair scheme with optimal access property.

We consider the decoding of linear and array codes from errors when we are only allowed to download a part of the codeword. More specifically, suppose that we have encoded k data symbols using an (n, k) code with code length n and dimension k. During storage, some of the codeword coordinates might be corrupted by errors. We aim to recover the original data by reading the corrupted codeword with a limit on the transmission bandwidth, namely, we can only download an α proportion of the corrupted codeword. For a given α, our objective is to design a code and a decoding scheme such that we can recover the original data from the largest possible number of errors. A naive scheme is to read αn coordinates of the codeword. This method used in conjunction with MDS codes guarantees recovery from any ⌊(αn - k)/2⌋ errors. In this paper we show that we can instead download an α proportion from each of the codeword's coordinates. For a well-designed MDS code, this method can guarantee recovery from ⌊(n - k/α)/2⌋ errors, which is 1/α times more than the naive method, and is also the maximum number of errors that an (n, k) code can correct by downloading only an α proportion of the codeword. We present two families of such optimal constructions and decoding schemes of which one is based on Interleaved Reed-Solomon codes and the other on Folded Reed-Solomon codes. We further show that both code constructions attain asymptotically optimal list decoding radius when downloading only a part of the corrupted codeword. We also construct an ensemble of random codes that with high probability approaches the upper bound on the number of correctable errors when the decoder downloads an α proportion of the corrupted codeword.

Erasure codes in large-scale storage systems allow recovery of data from a failed node. A recently developed class of codes, locally repairable codes (LRCs), offers tradeoffs between storage overhead and repair cost. LRCs facilitate efficient recovery scenarios by adding parity blocks to the system. However, these additional blocks may eventually increase the number of blocks that must be reconstructed. Existing LRCs differ in their use of the parity blocks, in their locality semantics, and in their parameter space. Thus, existing theoretical models cannot directly compare different LRCs to determine which code offers the best recovery performance, and at what cost. We perform the first systematic comparison of existing LRC approaches. We analyze Xorbas, Azure’s LRCs, and Optimal-LRCs in light of two new metrics: average degraded read cost and normalized repair cost. We show the tradeoff between these costs and the code’s fault tolerance, and that different approaches offer different choices in this tradeoff. Our experimental evaluation on a Ceph cluster further demonstrates the different effects of realistic system bottlenecks on the benefit from each LRC approach. Despite these differences, the normalized repair cost metric can reliably identify the LRC approach that would achieve the lowest repair cost in each setup.

We consider erasure coding for a model of storage that assumes that nodes are organized into equally sized groups, called racks, such that repairing failed nodes is limited only by inter-rack communication, while transmission of data within a rack does not contribute to the repair bandwidth. We present explicit families of MDS array codes that support recovery of a single failed node from any number of helper racks using the minimum possible amount of inter-rack communication. One of our constructions also has the additional property of low access. Finally, we present a construction of scalar Reed-Solomon codes that support optimal repair for the rack-oriented storage model.

We introduce a dynamical model of node repair in distributed storage systems wherein the storage nodes are subjected to failures according to independent Poisson processes. The main parameter that we study is the time-average capacity of the network in the scenario where a fixed subset of the nodes support a higher repair bandwidth than the other nodes. The sequence of node failures generates random permutations of the nodes in the encoded block, and we model the state of the network as a Markov random walk on permutations of n elements. As our main result we show that the capacity of the network can be increased compared to the static (worst-case) model of the storage system, while maintaining the same (average) repair bandwidth, and we derive estimates of the increase. We also quantify the capacity increase in the case that the repair center has information about the sequence of the recently failed storage nodes.

Locally recoverable (LRC) codes provide ways of recovering erased coordinates of the codeword without having to access each of the remaining coordinates. A subfamily of LRC codes with hierarchical locality (H-LRC codes) provides added flexibility to the construction by introducing several tiers of recoverability for correcting different numbers of erasures. We present a general construction of codes with 2-level hierarchical locality from maps between algebraic curves and specialize it to several code families obtained from quotients of curves by a subgroup of the automorphism group, including rational, elliptic, Kummer, and Artin-Schreier curves. We further address the question of H-LRC codes with availability, and suggest a general construction of such codes from fiber products of curves. Detailed calculations of parameters for H-LRC codes with availability are performed for Reed-Solomon- and Hermitian-like code families. Finally, we construct asymptotically good families of H-LRC codes from curves related to the Garcia-Stichtenoth tower.

The repair problem in distributed storage addresses recovery of the data encoded using an erasure code, for instance, a Reed-Solomon (RS) code. We consider the problem of repairing a single node or multiple nodes in RS-coded storage systems using the smallest possible amount of inter-nodal communication. According to the cut-set bound, communication cost of repairing h ≥ 1 failed nodes for an (n, k = n - r) maximum distance separable (MDS) code using d helper nodes is at least dhl/(d + h - k), where l is the size of the node. Guruswami and Wootters (2016) initiated the study of efficient repair of RS codes, showing that they can be repaired using a smaller bandwidth than under the trivial approach. At the same time, their work as well as follow-up papers stopped short of constructing RS codes (or any scalar MDS codes) that meet the cut-set bound with equality. In this paper, we construct the families of RS codes that achieve the cut-set bound for repair of one or several nodes. In the single-node case, we present the RS codes of length n over the field F(ql), l = exp((1 + o(1))n logn) that meet the cut-set bound. We also prove an almost matching lower bound on l, showing that super-exponential scaling is both necessary and sufficient for scalar MDS codes to achieve the cut-set bound using linear repair schemes. For the case of multiple nodes, we construct a family of RS codes that achieve the cut-set bound universally for the repair of any h = 1,2, . . ., r failed nodes from any subset of d helper nodes, k ≤ d ≤ n - h. For a fixed number of parities r, the node size of the constructed codes is close to the smallest possible node size for codes with such properties.

Two widely studied models of multiple-node repair in distributed storage systems are centralized repair and cooperative repair. The centralized model assumes that all the failed nodes are recreated in one location, while the cooperative one stipulates that the failed nodes may communicate but are distinct, and the amount of data exchanged between them is included in the repair bandwidth. As our first result, we prove a lower bound on the minimum bandwidth of cooperative repair. We also show that the cooperative model is stronger than the centralized one, in the sense that any maximum distance separable (MDS) code with optimal repair bandwidth under the former model also has optimal bandwidth under the latter one. These results were previously known under the additional “uniform download” assumption, which is removed in our proofs. As our main result, we give explicit constructions of MDS codes with optimal cooperative repair for all possible parameters. More precisely, given any n, k, h, d such that 2 ≤ h ≤ n- d ≤ n- k we construct (n, k) MDS codes over the field F of size |F| ≥ (d + 1 - k)n that can optimally repair any h erasures from any d helper nodes. The repair scheme of our codes involves two rounds of communication. In the first round, each failed node downloads information from the helper nodes, and in the second one, each failed node downloads additional information from the other failed nodes. This implies that our codes achieve the optimal repair bandwidth using the smallest possible number of rounds.

Two widely studied models of multiple-node repair in distributed storage systems are centralized repair and cooperative repair. The centralized model assumes that all the failed nodes are recreated in one location, while the cooperative one stipulates that the failed nodes may communicate but are distinct, and the amount of data exchanged between them is included in the repair bandwidth. As our first result, we prove a lower bound on the minimum bandwidth of cooperative repair. We also show that the cooperative model is stronger than the centralized one, in the sense that MDS codes with optimal repair bandwidth under the former model have the same property under the latter one. These results were known under the assumption of uniform download which is removed in our proofs. As our main result, we give explicit constructions of MDS codes with optimal cooperative repair for all possible parameters. More precisely, given any n, k, h, d such that 2\leqslant h\leqslant n-d\leqslant n-k we construct (n, k) MDS codes over the field F of size |F|\geqslant(d+1-k)n that can optimally repair any h erasures from any d helper nodes. The repair scheme of our codes involves two rounds of communication. In the first round, each failed node downloads information from the helper nodes, and in the second one, each failed node downloads additional information from the other failed nodes. This implies that our codes achieve the optimal repair bandwidth using the smallest possible number of rounds.

Locally recoverable (LRC) codes provide ways of recovering erased coordinates of the codeword without having to access each of the remaining coordinates. A subfamily of LRC codes with hierarchical locality (H-LRC codes) provides added flexibility to the construction by introducing several tiers of recoverability for correcting different numbers of erasures. We present a general construction of codes on algebraic curves that have 2-level hierarchical locality and give a number of examples of such code families.

In this paper, we show how the presence of locality within a binary cyclic code can be exploited to improve decoding performance and to reduce decoding complexity. We pursue two approaches. Under the first approach, we show how the ordered statistics decoding (OSD) method can be modified by inserting a simple single round belief-propagation step at the start that involves only the local codes. The resultant locality-aware OSD algorithm yields an appreciable signal-to-noise ratio (SNR) gain for a given level of reliability and essentially the same level of decoder complexity. Under the second, trellis decoding approach, we show that the careful introduction of locality results in the creation of a cyclic subcode that possesses lower maximum state complexity. In addition, we present a simple means of deriving an upper bound to the state complexity profile of any cyclic code that is based only on the zeros of the code. Furthermore, we show how the decoding speed of either locality-aware OSD or trellis decoding can be significantly increased in the presence of locality, in the moderate-to-high SNR regime, by making the use of a quick-look decoder that often returns the maximum likelihood code word.

Locally recoverable (LRC) codes have recently been a focus point of research in coding theory due to their theoretical appeal and applications in distributed storage systems. In an LRC code, any erased symbol of a codeword can be recovered by accessing only a small number of other symbols. For LRC codes over a small alphabet (such as binary), the optimal rate-distance trade-off is unknown. We present several new combinatorial bounds on LRC codes including the locality-aware sphere packing and Plotkin bounds. We also develop an approach to linear programming (LP) bounds on LRC codes. The resulting LP bound gives better estimates in examples than the other upper bounds known in the literature. Further, we provide the tightest known upper bound on the rate of linear LRC codes with a given relative distance, an improvement over the previous best known bounds.

We consider the minimax estimation problem of a discrete distribution with support size k under privacy constraints. A privatization scheme is applied to each raw sample independently, and we need to estimate the distribution of the raw samples from the privatized samples. A positive number ∈ measures the privacy level of a privatization scheme. For a given ∈, we consider the problem of constructing optimal privatization schemes with ∈-privacy level, i.e., schemes that minimize the expected estimation loss for the worst-case distribution. Two schemes known in the literature provide order optimal performance in the high privacy regime where E is very close to 0, and in the low privacy regime where e∈≈ k, respectively. In this paper, we propose a new family of schemes which substantially improve the performance of the existing schemes in the medium privacy regime when 1 ≪ e∈≪ k. More concretely, we prove that when 3.822metric and by 30% under ℓ1metric over the existing schemes. We also prove a lower bound for the region e∈≪ k, which implies that our schemes are order optimal in this regime.

The repair problem for an (n, k) error-correcting code calls for recovery of an unavailable coordinate of the codeword by downloading as little information as possible from a subset of the remaining coordinates. Using the terminology motivated by coding in distributed storage, we attempt to repair a failed node by accessing information stored on d helper nodes, where k ≤ d ≤ n - 1, and using as little repair bandwidth as possible to recover the lost information. By the so-called cut-set bound (Dimakis et al., 2010), the repair bandwidth of an (n,k = n - r) MDS code using d helper nodes is at least dl/(d + 1 - k), where l is the size of the node. A number of constructions of MDS array codes have been shown to meet this bound with equality. In a related but separate line of work, Guruswami and Wootters (2016) studied repair of Reed-Solomon (RS) codes, showing that it is possible to perform repair using a smaller bandwidth than under the trivial approach. At the same time, their work as well as follow-up papers stopped short of constructing RS codes (or any scalar MDS codes) that meet the cut-set bound with equality, which has been an open problem in coding theory. In this work we present a solution to this problem, constructing RS codes of length n over the field of size ql, l = exp((1 + o(1))n log n) that meet the cut-set bound. We also prove an almost matching lower bound on l, showing that super-exponential scaling is both necessary and sufficient for achieving the cut-set bound using linear repair schemes. More precisely, we prove that for scalar MDS codes (including the RS codes) to meet this bound, the sub-packetization l must satisfy l ≥ exp((1 + o(1))k log k).

In this paper, we study the impact of locality on the decoding of binary cyclic codes under two approaches, namely ordered statistics decoding (OSD) and trellis decoding. Given a binary cyclic code having locality or availability, we suitably modify the OSD to obtain gains in terms of Signal-To-Noise ratio, for a given reliability and essentially the same level of decoder complexity. With regard to trellis decoding, we show that careful introduction of locality results in the creation of cyclic subcodes having lower maximum state complexity. We also present a simple upper-bounding technique on the state complexity profile, based on the zeros of the code. Finally, it is shown how the decoding speed can significantly be increased in the presence of locality, in the moderate-to-high SNR regime, by making use of a quick-look decoder that often returns the ML codeword.

We consider error correction by maximum distance separable (MDS) codes based on a part of the received codeword. Our problem is motivated by applications in distributed storage. While efficiently correcting erasures by MDS storage codes (the “repair problem”) has been widely studied in recent literature, the problem of correcting errors in a similar setting seems to represent a new question in coding theory. Suppose that k data symbols are encoded using an (n, k) MDS code, and some of the codeword coordinates are located on faulty storage nodes that introduce errors. We want to recover the original data from the corrupted codeword under the constraint that the decoder can download only an α proportion of the codeword (fractional decoding). For any (n, k) code we show that the number of correctable errors under this constraint is bounded above by ⌊(n - k/α)/2⌋. Moreover, we present two families of MDS array codes which achieves this bound with equality under a simple decoding procedure. The decoder downloads an α proportion of each of the codeword's coordinates, and provides a much larger decoding radius compared to the naive approach of reading some an coordinates of the codeword. One of the code families is formed of Reed-Solomon (RS) codes with well-chosen evaluation points, while the other is based on folded RS codes. Finally, we show that folded RS codes also have the optimal list decoding radius under the fractional decoding constraint.

We consider the minimax estimation problem of a discrete distribution with support size k under privacy constraints. A privatization scheme is applied to each raw sample independently, and we need to estimate the distribution of the raw samples from the privatized samples. A positive number ϵ measures the privacy level of a privatization scheme. For a given ϵ, we want to find the optimal privatization scheme which minimizes the expected estimation loss for the worst-case distribution. Two schemes in the literature provide order optimal performance in the high-privacy regime when ϵ is very close to 0, and in the low-privacy regime when eϵ≈ k, respectively. In this paper, we propose a new family of schemes which substantially improve the performance of the existing schemes in the medium privacy regime when 1 ≪ eϵ≪ k. More concretely, we prove that when 3.822metric and 30% under ℓ1metric over the existing schemes. We also prove a tight lower bound for the whole region eϵ≪ k, which implies that our schemes are order optimal in this regime.

In group testing, simple binary-output tests are designed to identify a small number t of defective items that are present in a large population of N items. Each test takes as input a group of items and produces a binary output indicating whether the group is free of the defective items or contains one or more of them. In this paper, we study a relaxation of the combinatorial group testing problem. A matrix is called (t, ε)-disjunct if it gives rise to a nonadaptive group testing scheme with the property of identifying a uniformly random t-set of defective subjects out of a population of size N with false positive probability of an item at most ε. We establish a new connection between (t, ε)-disjunct matrices and error correcting codes based on the dual distance of the codes and derive estimates of the parameters of codes that give rise to such schemes. Our methods rely on the moments of the distance distribution of codes and inequalities for moments of sums of independent random variables. We also provide a new connection between group testing schemes and combinatorial designs.

A code over a finite alphabet is called locally recoverable (LRC code) if every symbol in the encoding is a function of a small number (at most r) of other symbols of the codeword. In this paper, we introduce a construction of LRC codes on algebraic curves, extending a recent construction of the Reed-Solomon like codes with locality. We treat the following situations: local recovery of a single erasure, local recovery of multiple erasures, and codes with several disjoint recovery sets for every coordinate (the availability problem). For each of these three problems we describe a general construction of codes on curves and construct several families of LRC codes. We also describe a construction of codes with availability that relies on automorphism groups of curves. We also consider the asymptotic problem for the parameters of the LRC codes on curves. We show that the codes obtained from asymptotically maximal curves (for instance, Garcia-Stichtenoth towers) improve upon the asymptotic versions of the Gilbert-Varshamov bound for LRC codes.

Maximum distance separable (MDS) codes are optimal error-correcting codes in the sense that they provide the maximum failure tolerance for a given number of parity nodes. Suppose that an MDS code with k information nodes and r = n - k parity nodes is used to encode data in a distributed storage system. It is known that if h out of the n nodes are inaccessible and d surviving (helper) nodes are used to recover the lost data, then we need to download at least h/(d + h - k) fraction of the data stored in each of the helper nodes (Dimakis et al., 2010 and Cadambe et al., 2013). If this lower bound is achieved for the repair of any h erased nodes from any d helper nodes, we say that the MDS code has the (h, d)-optimal repair property. We study high-rate MDS array codes with the optimal repair property (also known as minimum storage regenerating codes, or MSR codes). Explicit constructions of such codes in the literature are only available for the cases where there are at most three parity nodes, and these existing constructions can only optimally repair a single node failure by accessing all the surviving nodes. In this paper, given any r and n, we present two explicit constructions of MDS array codes with the (h, d)-optimal repair property for all h r and k d n - h simultaneously. Codes in the first family can be constructed over any base field F as long as |F| ≥ sn, where s = lcm(1, 2, . .. , r). The encoding, decoding, repair of failed nodes, and update procedures of these codes all have low complexity. Codes in the second family have the optimal access property and can be constructed over any base field F as long as |F| ≥ n+1. Moreover, both code families have the optimal error resilience capability when repairing failed nodes. We also construct several other related families of MDS codes with the optimal repair property.

An (n, k, l) maximum distance separable (MDS) array code of length n, dimension k = n - r, and subpacketization l is formed of l × n matrices over a finite field F, with every column of the matrix stored on a separate node in the distributed storage system and viewed as a coordinate of the codeword. Repair of a failed node (recovery of one erased column) can be performed by accessing a set of d ≤ n - 1 surviving (helper) nodes. The code is said to have the optimal access property if the amount of data accessed at each of the helper nodes meets a lower bound on this quantity. For optimal-access MDS codes with d = n-1, the sub-packetization l satisfies the bound l ≥ r(k-1)/r. In our previous work (IEEE Trans. Inf. Theory, vol. 63, no. 4, 2017), for any n and r, we presented an explicit construction of optimal-access MDS codes with subpacketization l = r⌈n-1⌉. In this paper, we take up the question of reducing the sub-packetization value l to make it to approach the lower bound. We construct an explicit family of optimal-access codes with l = r1.n/rl, which differs from the optimal value by at most a factor of r2. These codes can be constructed over any finite field F as long as |F| ≥ r⌈n/r⌉, and afford low-complexity encoding and decoding procedures. We also define a version of the repair problem that bridges the context of regenerating codes and codes with locality constraints (LRC codes), which we call group repair with optimal access. In this variation, we assume that the set of n = sm nodes is partitioned into m repair groups of size s, and require that the amount of accessed data for repair is the smallest possible whenever the d = s + k - 1 helper nodes include all the other s-1 nodes from the same group as the failed node. For this problem, we construct a family of codes with the group optimal access property. These codes can be constructed over any field F of size |F| ≥ n, and also afford low-complexity encoding and decoding procedures.

It is known that polar codes can be efficiently constructed for binary-input channels. At the same time, existing algorithms for general input alphabets are less practical because of high complexity. We address the construction problem for the general case, and analyze an algorithm that is based on successive reduction of the output alphabet size of the subchannels in each recursion step. For this procedure, we estimate the approximation error as O(μ-1/(q-1)), where q is the input alphabet size and μ is the “quantization parameter,” i.e., the maximum size of the subchannel output alphabet allowed by the algorithm. The complexity of the code construction scales as O(N μ2log μ), where N is the length of the code. We also show that if the polarizing operation relies on modulo-q addition, it is possible to merge subsets of output symbols without any loss in subchannel capacity. Performing this procedure before each approximation step results in a further speed-up of the code construction, and the resulting codes have smaller gap to capacity. We also show that a similar acceleration can be attained for polar codes over finite field alphabets. Experimentation shows that the suggested construction algorithms can be used to construct long polar codes for alphabets of size q = 16 and more with acceptable loss of the code rate for a variety of polarizing transforms.

A locally recoverable (LRC) code is a code that enables a simple recovery of an erased symbol by accessing only a small number of other symbols. We present several new combinatorial bounds on LRC codes including the locality-aware sphere packing and Plotkin bounds. We also develop an approach to linear programming (LP) bounds on LRC codes. The resulting LP bound gives better estimates in examples than the other upper bounds known in the literature.

Despite a large volume of research in group testing, explicit small-size group testing schemes are still difficult to construct, and the parameters of known combinatorial schemes are limited by the constraints of the problem. Relaxing the worst-case identification requirements to probabilistic localization of defectives enables one to expand the range of parameters, and yet the small-size practical constructions are sparse. Motivated by this question, we perform an experimental study of almost disjunct matrices constructed from low-weight codewords of binary BCH codes, and evaluate their performance in nonadaptive group testing. We observe that identification of defectives is much more stable in these schemes compared to the schemes constructed from random binary matrices. We derive an estimate of the error probability of identification in the constructed schemes which provides a partial explanation of their performance.

Given any r and n, we present an explicit construction of high-rate maximum distance separable (MDS) array codes that can optimally repair any d failed nodes from any h helper nodes for all h, 1 ≤ h ≤ r and d, k ≤ d ≤ n - h simultaneously. These codes can be constructed over any base field F as long as |F| ≥ sn; where s = lcm(1, 2,..., r). The encoding, decoding, repair of failed nodes, and update procedures of these codes all have low complexity. Our results present a significant improvement over earlier results which can only construct explicit codes for the case of at most 3 parity nodes, and these existing constructions can only optimally repair a single node failure by accessing all the surviving nodes. In the second part of the paper we give an explicit construction of Reed-Solomon codes with asymptotically optimal repair bandwidth.

A locally recoverable code (LRC code) is a code over a finite alphabet, such that every symbol in the encoding is a function of a small number of other symbols that form a recovering set. In this paper, we derive new finite-length and asymptotic bounds on the parameters of LRC codes. For LRC codes with a single recovering set for every coordinate, we derive an asymptotic Gilbert-Varshamov type bound for LRC codes and find the maximum attainable relative distance of asymptotically good LRC codes. Similar results are established for LRC codes with two disjoint recovering sets for every coordinate. For the case of multiple recovering sets (the availability problem), we derive a lower bound on the parameters using expander graph arguments. Finally, we also derive finite-length upper bounds on the rate and the distance of LRC codes with multiple recovering sets.

A code over a finite alphabet is called locally recoverable (LRC code) if every symbol in the encoding is a function of a small number (at most r) other symbols. A family of linear LRC codes that generalize the classic construction of Reed-Solomon codes was constructed in a recent paper by I. Tamo and A. Barg (IEEE Trans. Inform. Theory, vol. 60, no. 8, 2014, pp. 4661-4676). In this paper we extend this construction to codes on algebraic curves. We give a general construction of LRC codes on curves and compute some examples, including asymptotically good families of codes derived from the Garcia-Stichtenoth towers. The local recovery procedure is performed by polynomial interpolation over r coordinates of the codevector. We also obtain a family of Hermitian codes with two disjoint recovering sets for every symbol of the codeword.

We consider linear cyclic codes with the locality property, or locally recoverable codes (LRC codes). A family of LRC codes that generalizes the classical construction of Reed-Solomon codes was constructed in a recent paper by I. Tamo and A. Barg (IEEE Trans. IT, no. 8, 2014). In this paper we focus on the optimal cyclic codes that arise from the general construction. We give a characterization of these codes in terms of their zeros, and observe that there are many equivalent ways of constructing optimal cyclic LRC codes over a given field. We also study subfield subcodes of cyclic LRC codes (BCH-like LRC codes) and establish several results about their locality and minimum distance.

In the standard polar coding construction, the same kernel is used for every channel transformation. In this paper we propose a new polar coding scheme in which the channel combining transformation is dynamically chosen based on the properties of the virtual channels. We show that for some cases it can improve the finite-length performance of codes. We also show that the dynamic scheme retains some of the main properties of the original polar code construction.

The wiretap channel model of Wyner is one of the first communication models with both reliability and security constraints. Capacity-achieving schemes for various models of the wiretap channel have received considerable attention in recent literature. In this paper, we show that capacity of the general (not necessarily degraded or symmetric) wiretap channel under a “strong secrecy constraint” can be achieved using a transmission scheme based on polar codes. We also extend our construction to the case of broadcast channels with confidential messages defined by Csiszár and Körner, achieving the entire capacity region of this communication model.

We study statistical restricted isometry, a property closely related to sparse signal recovery, of deterministic sensing matrices of size m × N. A matrix is said to have a statistical restricted isometry property (StRIP) of order k if most submatrices with k columns define a near-isometric map of Rk into Rm. As our main result, we establish sufficient conditions for the StRIP property of a matrix in terms of the mutual coherence and mean square coherence. We show that for many existing deterministic families of sampling matrices, m = O(k) rows suffice for k-StRIP, which is an improvement over the known estimates of either m = Θ(k log N) or m = Θ(k log k). We also give examples of matrix families that are shown to have the StRIP property using our sufficient conditions.

A code over a finite alphabet is called locally recoverable code (LRC code) if every symbol in the encoding is a function of a small number (at most r) other symbols. We present a family of LRC codes that attain the maximum possible value of the distance for a given locality parameter and code cardinality. The codes can be constructed over a finite field alphabet of any size that exceeds the code length. The codewords are obtained as evaluations of specially constructed polynomials over a finite field, and reduce to Reed-Solomon codes if the locality parameter r is set to be equal to the code dimension. The recovery procedure is performed by polynomial interpolation over r points. We also construct codes with several disjoint recovering sets for every symbol. This construction enables the system to conduct several independent and simultaneous recovery processes of a specific symbol by accessing different parts of the codeword. This property enables high availability of frequently accessed data.

A locally recoverable code (LRC code) is a code over a finite alphabet such that every symbol in the encoding is a function of a small number of other symbols that form a recovering set. Bounds on the rate and distance of such codes have been extensively studied in the literature. In this paper we derive upper bounds on the rate and distance of codes in which every symbol has t ≥ 1 disjoint recovering sets.

A code over a finite alphabet is called locally recoverable (LRC) if every symbol in the encoding is a function of a small number (at mostr) other symbols. We present a family of LRC codes that attain the maximum possible value of the distance for a given locality parameter and code cardinality. The codewords are obtained as evaluations of specially constructed polynomials over a finite field, and reduce to a Reed-Solomon code if the locality parameterris set to be equal to the code dimension. The size of the code alphabet for most parameters is only slightly greater than the code length. The recovery procedure is performed by polynomial interpolation overrpoints. We also construct codes with several disjoint recovering sets for every symbol. This construction enables the system to conduct several independent and simultaneous recovery processes of a specific symbol by accessing different parts of the codeword. This property enables high availability of frequently accessed data (“hot data”).

We design polar codes for q-ary input, q = 2r, that polarize to an arbitrary given subset of extremal configurations out of the original triangular array of r +1 such configurations. We also obtain a new proof of the known result about 2-level polarization of q-ary codes.

Ann-wordy=(y1,...,yn) over a finite alphabet of cardinalityqis called a descendant of a set oftwordsx1,...,xtif every coordinateyi,i=1,...,n, is contained in the set {x1i,...,xti}. A codeC={x1,...,xM} is said to have thet-IPP property if for anyn-wordythat is a descendant of at mosttparents belonging to the code, it is possible to identify at least one of them. From earlier works, it is known thatt-IPP codes of positive rate exist if and only ift≤q-1. We introduce a robust version of IPP codes which allows error-free identification of parents in the presence of a certain number of mutations, i.e., coordinates inythat can break away from the descent rule, taking arbitrary values from the alphabet or becoming completely unreadable. We show existence of robustt-IPP codes for allt≤q-1 and some positive proportion of such coordinates. We uncover a relation between the hash distance of codes and the IPP property and use it to find the exact proportion of mutant coordinates that permits identification of pirates with zero probability of error in the case of size-2 coalitions.

Rank modulation is a way of encoding information to correct errors in flash memory devices as well as impulse noise in transmission lines. Modeling rank modulation involves construction of packings of the space of permutations equipped with the Kendall tau distance. As our main set of results, we present several general constructions of codes in permutations that cover a broad range of code parameters. In particular, we show a number of ways in which conventional error-correcting codes can be modified to correct errors in the Kendall space. Our constructions are nonasymptotic and afford simple encoding and decoding algorithms of essentially the same complexity as required to correct errors in the Hamming metric. As an example, from binary Bose–Chaudhuri–Hocquenghem codes, we obtain codes correcting$t$Kendall errors in$n$memory cells that support the order of$n!/(\log_{2}n!)^{t}$messages, for any constant$t=1,2,\ldots$. We give many examples of rank modulation codes with specific parameters. Turning to asymptotic analysis, we construct families of rank modulation codes that correct a number of errors that grows with$n$at varying rates, from$\Theta (n)$to$\Theta (n^{2})$. One of our constructions gives rise to a family of rank modulation codes for which the tradeoff between the number of messages and the number of correctable Kendall errors approaches the optimal scaling rate.

We study polarization for nonbinary channels with input alphabet of sizeq=2r,r=2,3,.... Using Arıkan's polarizing kernelH2, we prove that the virtual channels that arise in the process of channel evolution converge toq-ary channels with capacity 0,1,2,...,rbits. As a result of this analysis, we show that polar codes support reliable transmission over discrete memoryless channels withq-ary input for all rates below the symmetric capacity of the channel. This leads to an explicit transmission scheme forq-ary channels. The block error probability of decoding using successive cancellation behaves as exp(-Nα), whereNis the code length and α is any constant less than 0.5.

Poset metrics form a generalization of the Hamming metric on the space Fnq. Orbits of the group of linear isometries of the space give rise to a translation association scheme. The structure of the dual scheme is important in studying duality of linear codes; this study is facilitated if the scheme is self-dual. We study the relation between self-duality of the scheme and that of the poset. We also give new examples of poset metric spaces and describe the association schemes that arise from linear isometries.

We study polarization for nonbinary channels with input alphabet of size q = 2r, r = 2,3,.... Using Arikan's polarizing kernel H2, we prove that virtual channels that arise in the process of polarization converge to q-ary channels with capacity 1,2,..., r bits, and that the total transmission rate approaches the symmetric capacity of the channel. This leads to an explicit transmission scheme for q-ary channels. The error probability of decoding using successive cancellation behaves as exp(-Nα), where N is the code length and α is any constant less than 0.5.

We introduce a MinSum-based list decoder for product codes and analyze its performance. We show that it can guarantee successful list decoding for decoding radii that lie above half the code's minimum distance.

Compressive sampling is a technique of recovering sparse N-dimensional signals from low-dimensional sketches, i.e., their linear images in ℝm, m ≪ N. The main question associated with this technique is construction of linear operators that allow faithful recovery of the signal from its sketch. The most frequently used sufficient condition for robust recovery is the near-isometry property of the operator when restricted to k-sparse signals. We study ±1-matrices of dimensions m × N that satisfy the restricted isometry property of order k (k-RIP). As our main set of results, we describe a general method of constructing sampling matrices for which a statistical version of k-RIP holds. We also show that m×N matrices with k-RIP and m = O(k2logN) can be constructed with time complexity O(k2N logN).

We study coding for binary channels in which out of any two consecutive transmitted bits at most one can be affected by errors. We consider a set of basic coding problems for such channels, deriving estimates on the size of optimal codes and providing some constructions. We also study a generalization to errors separated by at least s = 2, 3, ... error-free channel uses. Finally, we define a probabilistic model of a binary channel with non-adjacent errors and find the capacity of this channel.

Rank modulation is a way of encoding information to correct errors in flash memory devices as well as impulse noise in transmission lines. Modeling rank modulation involves construction of packings of the space of permutations equipped with the Kendall tau distance. We present several general constructions of codes in permutations that cover a broad range of code parameters. In particular, we show that a code that corrects Hamming errors can be used to construct a code for correcting Kendall errors. For instance, from BCH codes we obtain codes correcting t Kendall errors in n memory cells that support the order of n!/ logtn! messages, for any t = 1, 2, .... We also construct families of codes that correct a number of errors that grows with n at varying rates, from Θ(n) to Θ(n2).

The ordered Hamming metric is a generalization of the usual Hamming distance derived from a partial order on the set of coordinates. So far, coding theory in ordered spaces has primarily focused on combinatorial aspects. The main object of this paper is to develop the relation of the ordered Hamming space to the context of information transmission. Using the models in previous works of Rosenbloom and Tsfasman (1997) and Tavildar and Viswanath (2006) as a starting point, we define the ordered symmetric channel and the ordered erasure channel which are counterparts of the q-ary symmetric channel and the q-ary erasure channel, respectively. We establish a set of basic results for these channels as well as their relation to linear ordered codes.

We study ensembles of codes on graphs (generalized low-density parity-check, or LDPC codes) constructed from random graphs and fixed local constrained codes, and their extension to codes on hypergraphs. It is known that the average minimum distance of codes in these ensembles grows linearly with the code length. We show that these codes can correct a linearly growing number of errors under simple iterative decoding algorithms. In particular, we show that this property extends to codes constructed by parallel concatenation of Hamming codes and other codes with small minimum distance. Previously known results that proved this property for graph codes relied on graph expansion and required the choice of local codes with large distance relative to their length.

In terabit-density magnetic recording, several bits of data can be replaced by the values of their neighbors in the storage medium. As a result, errors in the medium are dependent on each other and also on the data written. We consider a simple 1-D combinatorial model of this medium. In our model, we assume a setting where binary data is sequentially written on the medium and a bit can erroneously change to the immediately preceding value. We derive several properties of codes that correct this type of errors, focusing on bounds on their cardinality. We also define a probabilistic finite-state channel model of the storage medium, and derive lower and upper estimates of its capacity. A lower bound is derived by evaluating the symmetric capacity of the channel, i.e., the maximum transmission rate under the assumption of the uniform input distribution of the channel. An upper bound is found by showing that the original channel is a stochastic degradation of another, related channel model whose capacity we can compute explicitly.

We develop the concept of hierarchical fingerprinting introduced in our recent work (ISIT2009). The object of two-level fingerprinting is content protection against coalitions of t pirates in such a manner that one of the pirates can be identified exactly if t ≤ t2or localized to within a small group if t21, where t1and t2are parameters of the system. In this work, we require the additional property that in the latter case no innocent users are accused of belonging to the pirate coalition. We construct two-level fingerprinting codes with polynomial complexity of identification that satisfy the strong definition of hierarchical fingerprinting described above.

Codes for rank modulation have been recently proposed as a means of protecting flash memory devices from errors. We study basic coding theoretic problems for such codes, representing them as subsets of the set of permutations of n elements equipped with the Kendall tau distance. We derive several lower and upper bounds on the size of codes. These bounds enable us to establish the exact scaling of the size of optimal codes for large values of n. We also show the existence of codes whose size is within a constant factor of the sphere packing bound for any fixed number of errors.

We prove a new bound on the size of codes with few distances in the Hamming space, improving an earlier result of P. Delsarte. We also improve the Ray-Chaudhuri-Wilson bound of the size of uniform intersecting families of subsets (constant-weight codes) and the bound of Delsarte-Goethals-Seidel on the maximum size of spherical codes with few distances. Finally, we find the size of maximal binary codes and maximal constant-weight codes of small length with 2,3, and 4 distances.

In the area of compressed sensing it is known that high-dimensional, approximately sparse signals x ∈ RNcan be accurately recovered from a small number m of linear samples. Most known recovery procedures rely in some form on a special property of the sampling matrix, called the Restricted Isometry Property (RIP). It has been shown that random Gaussian matrices possess the RIP and allow an efficient reconstruction of the signal, for instance, by linear programming or by other methods, and similar results are known for the ensembles of random binary matrices. We pursue a link between coding theory and compressed sensing, discussing the construction problem of sampling matrices with short sketches. The best known constructions of “small-bias probability spaces” (aka codes with narrow distance distribution) account for deterministic sampling matrices with sketch length m = O(k3log N/log k) while the best randomized constructions require only m = Ω(k log(N/k)) samples, where k is the number of essential entries of the signal. We address the construction problem of sampling matrices in the region between these two extremes, allowing the sketch length of order k2log N. We show that in this case it is possible to construct ensembles of sampling matrices of size much smaller than the ensembles known previously. For instance, the number of random bits required to construct a matrix in one of our ensembles equals O(k2log k logN), which compares favorably with the previously known estimate of O(kN log(N/k)) random bits for ensembles of random binary matrices.

We study q-ary codes with distance defined by a partial order of the coordinates of the codewords. Maximum Distance Separable (MDS) codes in the poset metric have been studied in a number of earlier works. We consider codes that are close to MDS codes by the value of their minimum distance. For such codes, we determine their weight distribution, and in the particular case of the “ordered metric” characterize distributions of points in the unit cube defined by the codes. We also give some constructions of codes in the ordered Hamming space.

We study a model of errors in binary data that arises in terabit-density magnetic recording. Under this model, several bits of the data are replaced by the values of their neighbors on the medium. We consider a simple one-dimensional version of this model, and derive several properties of codes that correct this type of error.

Codes with the identifiable parent property (IPP codes) are used in traitor tracing schemes that protect data broadcast by the publisher from unauthorized access or distribution. An n-word y over a finite alphabet is called a descendant of a set of t words x1, ..., xtif yiϵ {x1i, ..., xti} for all i = 1, ... n. A code C = {x1, ..., xM} is said to have the i-IPP property if for any n-word y that is a descendant of at most t parents belonging to the code it is possible to identify at least one of them. The existence of good i-IPP codes is known from earlier works. We introduce a robust version of IPP codes which allows unconditional identification of parents even if some of the coordinates in y can break away from the descent rule, i.e., can take arbitrary values from the alphabet, or become completely unreadable. By linking this problem to perfect hash functions and, more generally, to hash distances of a code, we prove initial results on the proportion of such coordinates that can be tolerated under the unconditional recovery requirement.

Codes for rank modulation have been recently proposed as a means of protecting flash memory devices from errors. We study basic coding theoretic problems for such codes, representing them as subsets of the set of permutations ofnelements equipped with the Kendall tau distance. We derive several lower and upper bounds on the size of codes. These bounds enable us to establish the exact scaling of the size of optimal codes for large values of n. We also show the existence of codes whose size is within a constant factor of the sphere packing bound for any fixed number of errors.

We consider secret key generation for a “pairwise independent network” model in which every pair of terminals observes correlated sources that are independent of sources observed by all other pairs of terminals. The terminals are then allowed to communicate publicly with all such communication being observed by all the terminals. The objective is to generate a secret key shared by a given subset of terminals at the largest rate possible, with the cooperation of any remaining terminals. Secrecy is required from an eavesdropper that has access to the public interterminal communication. A (single-letter) formula for secret key capacity brings out a natural connection between the problem of secret key generation and a combinatorial problem of maximal packing of Steiner trees in an associated multigraph. An explicit algorithm is proposed for secret key generation based on a maximal packing of Steiner trees in a multigraph; the corresponding maximum rate of Steiner tree packing is thus a lower bound for the secret key capacity. When only two of the terminals or when all the terminals seek to share a secret key, the mentioned algorithm achieves secret key capacity in which case the bound is tight.

We introduce the notion of two-level fingerprinting and traceability codes. In this setting, the users are organized in a hierarchical manner by classifying them into various groups; for instance, by dividing the distribution area into several geographic regions, and collecting users from the same region into one group. Two-level fingerprinting and traceability codes have the following property: As in traditional (one-level) codes, when given an illegal copy produced by a coalition of users, the decoder identifies one of the guilty users if the coalition size is less than a certain threshold t. Moreover, even when the coalition is of a larger size s (≫ t), the decoder still provides partial information by tracing one of the groups containing a guilty user. We establish sufficient conditions for a code to possess the two-level traceability property. In addition, we also provide constructions for two-level fingerprinting codes and characterize the corresponding set of achievable rates.

We estimate the number of errors corrected by two different ensembles of codes on graphs (generalized LDPC codes), namely codes on regular bipartite graphs and their extension to hypergraphs.

We investigate perfect secret key generation for a ¿pairwise independent network¿ model in which every pair of terminals observes correlated sources that are independent of sources observed by all other pairs of terminals. The terminals are then allowed to communicate interactively in multiple rounds over a public noiseless channel of unlimited capacity. This communication is observed by all the terminals as well as by an eavesdropper. The objective is to generate a perfect secret key shared by a given set of terminals at the largest rate possible. All the terminals cooperate in generating the secret key, with perfect secrecy being required from the eavesdropper. For this model, we introduce the concept of communication for perfect omniscience using which we first obtain a single-letter characterization of the perfect secret key capacity. Moreover, this perfect secret key capacity is shown to be achieved by linear noninteractive communication, and coincides with the (standard) secret key capacity. Our second contribution, exploiting the notion of communication for perfect omniscience, is a new nonasymptotic and computable upper bound for the combinatorial problem of maximal Steiner tree packing in a multigraph. Thus, our work establishes certain connections among perfect secrecy generation and communication for perfect omniscience for the pairwise independent network model, and Steiner tree packing.

A generalization of codes on regular bipartite graphs is given by a family of codes on hypergraphs. We derive the average weight distribution and estimate the minimum distance of codes in the random ensemble of hypergraph codes. We also propose an iterative decoding algorithm of hypergraph codes that corrects a larger proportion of errors than known previously for this code family.

We investigate secret key generation for a ldquopair-wise independent networkrdquo model in which every pair of terminals observes correlated sources which are independent of sources observed by all other pairs of terminals. The terminals are then allowed to communicate interactively in multiple rounds over a public noiseless channel of unlimited capacity, with all such communication being observed by all the terminals. The objective is to generate a secret key shared by a given subset of terminals at the largest rate possible. All the terminals cooperate in generating the secret key, with secrecy being required from an eavesdropper which has access to the public interterminal communication. We provide a (single-letter) formula for the secrecy capacity for this model, and show a natural connection between the problem of secret key generation and the combinatorial problem of maximal packing of Steiner trees in an associated multigraph. In particular, we show that the maximum number of Steiner tree packings in the multigraph is always a lower bound for the secrecy capacity. The bound is tight for the case when all the terminals seek to share a secret key; the mentioned connection yields an explicit capacity-achieving algorithm. This algorithm, which can be executed in polynomial time, extracts a group-wide secret key of the optimum rate from the collection of optimum and mutually independent secret keys for pairs of terminals.

We address the maximum attainable rate of fingerprinting codes under the marking assumption, studying lower and upper bounds on the value of the rate for various sizes of the attacker coalition. Lower bounds are obtained by considering typical coalitions, which represents a new idea in the area of fingerprinting and enables us to improve the previously known lower bounds for coalitions of size two and three. For upper bounds, the fingerprinting problem is modeled as a communications problem. It is shown that the maximum code rate is bounded above by the capacity of a certain class of channels, which are similar to the multiple-access channel (MAC). Converse coding theorems proved in the paper provide new upper bounds on fingerprinting capacity.

We investigate the decoding region for algebraic soft-decision decoding (ASD) of Reed-Solomon (RS) codes in a discrete, memoryless, additive-noise channel. An expression is derived for the error correction radius within which the soft-decision decoder produces a list that contains the transmitted codeword. The error radius for ASD is shown to be larger than that of Guruswami-Sudan (GS) hard-decision decoding for a subset of low and medium-rate codes. These results are also extended to multivariable interpolation in the sense of Parvaresh and Vardy.

We study the maximum attainable rate or capacity of fingerprinting codes under the marking assumption. It is proved that capacity for fingerprinting against coalitions of size two and three over the binary alphabet satisfies 0.25 ≤ C2,2≤ 0.322 and 0.083 ≤ C3,2≤ 0.199 respectively. For coalitions of an arbitrary fixed size, we derive a closed-form upper bound on fingerprinting capacity in the binary case. Finally, for general alphabets, we establish upper bounds on the fingerprinting capacity involving only single-letter mutual information quantities.

We prove several new bounds on ordered codes and ordered orthogonal arrays. We also show that the eigenvalues of the ordered Hamming scheme are the multivariable Krawtchouk polynomials and establish some of their properties.

We bound the probability of error of Algebraic Soft-Decision Decoding (ASD) of Reed-Solomon codes in a discrete, memoryless, additive-noise channel. First, we derive an exponential error bound on the probability of list-decoding error that improves on the bound presented in [9]. However, our previous work [1] has shown that list-decoding error is not sufficient to adequately characterize ASD's performance. Thus, we redefine an error as the event that the decoder selects an erroneous codeword from its list, and we derive an exponential error bound for the probability of selection error.

New lower bounds are established on the rate of fingerprinting codes secure against coalitions of an arbitrary constant size t. In particular, it is proved that there exist sequences of binary fingerprinting codes with vanishing probability of misidentification and rate approximately 1/(t 2t). In the case of t = 3 it is shown that there exist codes of rate 0.064 which is better by an order of magnitude than previously known results.

We derive a new upper bound on the size of a code in the Grassmannian space. The bound is asymptotically better than the upper bounds known previously in the entire range of distances except very large values

The minimum distance of some families of expander codes is studied, as well as some related families of codes defined on bipartite graphs. The weight spectrum and the minimum distance of a random ensemble of such codes are computed and it is shown that it sometimes meets the Gilbert-Varshamov (GV) bound. A lower bound on the minimum distances of constructive families of expander codes is derived. The relative minimum distance of the expander code is shown to exceed the product bound, i.e., the quantity /spl delta//sub 0//spl delta//sub 1/ where /spl delta//sub 0/ and /spl delta//sub 1/ are the minimum relative distances of the constituent codes. As a consequence of this, a polynomially constructible family of expander codes is obtained whose relative distance exceeds the Zyablov bound on the distance of serial concatenations.

We define multilevel codes on bipartite graphs which have properties analogous to multilevel serial concatenations. A linear-time decoding algorithm is described that corrects a proportion of errors equal to half the Blokh-Zyablov bound. The error probability of this algorithm has exponent similar to that of serially concatenated multilevel codes, i.e. equals the best-known exponent achievable by a polynomial-time decoding algorithm

We address the problem of bounding below the probability of error under maximum-likelihood decoding of a binary code with a known distance distribution used on a binary-symmetric channel (BSC). An improved upper bound is given for the maximum attainable exponent of this probability (the reliability function of the channel). In particular, we prove that the "random coding exponent" is the true value of the channel reliability for codes rate R in some interval immediately below the critical rate of the channel. An analogous result is obtained for the Gaussian channel.

In the above titled paper (ibid., vol. 48, no. 9, pp.2450-2454, Sep 02), corrections were made to Haar measurements in various equations.

An analogy is examined between serially concatenated codes and parallel concatenations whose interleavers are described by bipartite graphs with good expanding properties. In particular, a modified expander code construction is shown to behave very much like Forney's classical concatenated codes, though with improved decoding complexity. It is proved that these new codes achieve the Zyablov bound /spl delta//sub Z/ on the minimum distance. For these codes, a soft-decision, reliability-based, linear-time decoding algorithm is introduced, that corrects any fraction of errors up to almost /spl delta//sub Z//2. For the binary-symmetric channel, this algorithm's error exponent attains the Forney bound previously known only for classical (serial) concatenations.

An improved bound is proved on the list-decoding radius of a concatenated code relying upon a combination of (soft-decision) algebraic list decoding and generalized minimum distance (GMD) decoding in the outer level. This bound is further improved if the inner code is a random linear code.

A constructive family of expander codes is presented whose minimum distance exceeds the product (Zyablov) bound for all code rates between 0 and 1. Weight spectrum and the minimum distance of a random ensemble of bipartite-graph codes are computed. It is shown that if the vertex codes have minimum distance /spl ges/3, the overall code is asymptotically good, and sometimes meets the Gilbert-Varshamov bound.

A class of codes is said to reach capacity {\scriptsize $Ç$} of the binary symmetric channel if for any rate $R < $ {\scriptsize $Ç$} and any $\varepsilon > 0$ there is a sufficiently large N such that codes of length $\ge N$ and rate R from this class provide error probability of decoding at most $\varepsilon$, under some decoding algorithm. The study of the error probability of expander codes was initiated by Barg and Z{émor in 2002 [IEEE Trans. Inform. Theory, 48 (2002), pp. 1725--1729], where it was shown that they attain capacity of the binary symmetric channel under a linear-time iterative decoding with error probability falling exponentially with code length N. In this work we study variations on the expander code construction and focus on the most important region of code rates, close to the channel capacity. For this region we estimate the decrease rate (the error exponent) of the error probability of decoding for randomized ensembles of codes. The resulting estimate gives a substantial improvement of previous results for expander codes and some other explicit code families.

We derive improved bounds on the error and erasure rate for spherical codes and for binary linear codes under Forney's erasure/list decoding scheme and prove some related results.

We consider a general fingerprinting problem of digital data under which coalitions of users can alter or erase some bits in their copies in order to create an illegal copy. Each user is assigned a fingerprint which is a word in a fingerprinting code of size M (the total number of users) and length n. We present binary fingerprinting codes secure against size-t coalitions which enable the distributor (decoder) to recover at least one of the users from the coalition with probability of error exp(-/spl Omega/(n)) for M=exp(/spl Omega/(n)). This is an improvement over the best known schemes that provide the error probability no better than exp(-/spl Omega/(n/sup 1/2/)) and for this probability support at most exp(O(n/sup 1/2/)) users. The construction complexity of codes is polynomial in n. We also present versions of these constructions that afford identification algorithms of complexity poly(n)=polylog(M), improving over the best previously known complexity of /spl Omega/(M). For the case t=2, we construct codes of exponential size with even stronger performance, namely, for which the distributor can either recover both users from the coalition with probability 1-exp(/spl Omega/(n)), or identify one traitor with probability 1.

A linear code can be thought of as a vector matroid represented by the columns of the code's generator matrix; a well-known result in this context is Greene's theorem on a connection of the weight polynomial of the code and the Tutte polynomial of the matroid. We examine this connection from the coding-theoretic viewpoint, building upon the rank polynomial of the code. This enables us to obtain bounds on all-terminal reliability of linear matroids and new proofs of two known results: Greene's theorem and a connection between the weight polynomial and the partition polynomial of the Potts model.

We extend a low-rate improvement of the random coding bound on the reliability of a classical discrete memoryless channel (DMC) to its quantum counterpart. The key observation that we make is that the problem of bounding below the error exponent for a quantum channel relying on the class of stabilizer codes is equivalent to the problem of deriving error exponents for a certain symmetric classical channel.

Minimum distances, distance distributions, and error exponents on a binary-symmetric channel (BSC) are given for typical codes from Shannon's random code ensemble and for typical codes from a random linear code ensemble. A typical random code of length N and rate R is shown to have minimum distance N/spl delta//sub GV/(2R), where /spl delta//sub GV/(R) is the Gilbert-Varshamov (GV) relative distance at rate R, whereas a typical linear code (TLC) has minimum distance N/spl delta//sub GV/(R). Consequently, a TLC has a better error exponent on a BSC at low rates, namely, the expurgated error exponent.

We derive the Gilbert-Varshamov and Hamming bounds for packings of spheres (codes) in the Grassmann manifolds over R and C. Asymptotic expressions are obtained for the geodesic metric and projection Frobenius (chordal) metric on the manifold.

We show that expander codes attain the capacity of the binary-symmetric channel under iterative decoding. The error probability has a positive exponent for all rates between zero and the channel capacity. The decoding complexity grows linearly with the code length.

Let C be a code of length n over an alphabet of q letters. An n-word y is called a descendant of a set of t codewords x 1 , . . . ,x t if $y_i\in\{x^1_i,\dots,x^t_i\}$ for all i=1, . . . ,n. A code is said to have the t-identifying parent property if for any n-word that is a descendant of at most t parents it is possible to identify at least one of them. We prove that for any $t\le q-1$ there exist sequences of such codes with asymptotically nonvanishing rate.

We consider the problem of bounding the distance distribution for unrestricted block codes with known distance and/or dual distance. Applying the polynomial method, we provide a general framework for previously known results. We derive several upper and lower bounds both for finite length and for sequences of codes of growing length. Asymptotic results in the paper improve previously known estimates. In particular, we prove the best known bounds on the binomiality range of the distance spectrum of codes with a known dual distance.

We derive lower bounds on the distance and error exponent of the coding scheme described in the title. The bounds are compared to the parameters and error performance of a concatenated code family with varying inner codes of equal rates and a fixed minimum-distance separable (MDS) code as the outer code, letting the inner and outer code lengths approach infinity.

This paper is devoted to the problem of error detection with quantum codes. We show that it is possible to give a consistent definition of the undetected error event. To prove this, we examine possible problem settings for quantum error detection. Our goal is to derive a functional that describes the probability of undetected error under natural physical assumptions concerning transmission with error detection with quantum codes. We discuss possible transmission protocols with stabilizer and unrestricted quantum codes. The set of results proved in the paper shows that in all the cases considered the average probability of undetected error for a given code is essentially given by one and the same function of its weight enumerators. We examine polynomial invariants of quantum codes and show that coefficients of Rains's (see ibid., vol44, p.1388-94, 1998) "unitary weight enumerators" are known for classical codes under the name of binomial moments of the distance distribution. As in the classical situation, these enumerators provide an alternative expression for the probability of undetected error.

In Part I of this paper we formulated the problem of error detection with quantum codes on the completely depolarized channel and gave an expression for the probability of undetected error via the weight enumerators of the code.In this part we show that there exist quantum codes whose probability of undetected error falls exponentially with the length of the code and derive bounds on this exponent.The lower (existence) bound is proved for stabilizer codes by the counting argument for classical self-orthogonal quaternary codes.Upper bounds are proved by linear programming.First we formulate two linear programming problems that are convenient for the analysis of specific short codes.Next we give a relaxed formulation of the problem in terms of optimization on the cone of polynomials in the Krawtchouk basis.We present two general solutions of the problem.Together they give an upper bound on the exponent of undetected error.The upper and lower asymptotic bounds coincide for a certain interval of code rates close to 1.

We derive a new upper bound on the exponent of error probability of decoding for the best possible codes in the Gaussian channel. This bound is tighter than the known upper bounds (the sphere-packing and minimum-distance bounds proved in Shannon's classical 1959 paper and their low-rate improvement by Kabatiansky and Levenshtein (1978)). The proof is accomplished by studying asymptotic properties of codes on the sphere S/sup n-1/(R). First we prove a general lower bound on the distance distribution of codes of large size. To derive specific estimates of the distance distribution, we study the asymptotic behavior of Jacobi polynomials P/sub k//sup ak, bk/ as k/spl rarr//spl infin/. Since on the average there are many code vectors in the vicinity of the transmitted vector x, one can show that the probability of confusing x and one of these vectors cannot be too small. This proves a lower bound on the error probability of decoding and the upper bound announced in the title.

We study a combinatorial invariant of codes which counts the number of ordered pairs of codewords in all subcodes of restricted support in a code. This invariant can be expressed as a linear form of the components of the distance distribution of the code with binomial numbers as coefficients. For this reason we call it a binomial moment of the distance distribution. Binomial moments appear in the proof of the MacWilliams (1963) identities and in many other problems of combinatorial coding theory. We introduce a linear programming problem for bounding these linear forms from below. It turns out that some known codes (1-error-correcting perfect codes, Golay codes, Nordstrom-Robinson code, etc.) yield optimal solutions of this problem, i.e., have minimal possible binomial moments of the distance distribution. We derive several general feasible solutions of this problem, which give lower bounds on the binomial moments of codes with given parameters, and derive the corresponding asymptotic bounds. Applications of these bounds include new lower bounds on the probability of undetected error for binary codes used over the binary-symmetric channel with crossover probability p and optimality of many codes for error detection. Asymptotic analysis of the bounds enables us to extend the range of code rates in which the upper bound on the undetected error exponent is tight.

We derive new asymptotic upper bounds on the generalized weights of a binary linear code of a given size. We also prove some asymptotic results on the distance distribution of binary codes.

We suggest a decoding algorithm of q-ary linear codes, which we call supercode decoding. It ensures the error probability that approaches the error probability of minimum-distance decoding as the length of the code grows. For n/spl rarr//spl infin/ the algorithm has the maximum-likelihood performance. The asymptotic complexity of supercode decoding is exponentially smaller than the complexity of all other methods known. The algorithm develops the ideas of covering-set decoding and split syndrome decoding.

We consider a communication model over a binary channel in which the transmitter knows which bits of the n-bit transmission are prone to loss in the channel. We call this model channel with localized erasures in analogy with localized errors studied earlier in the literature. We present two constructions of binary codes with t(1+/spl epsiv/) check bits, where t=/spl alpha/n is the maximal possible number of erasures. One construction is on-line and has encoding complexity of order n//spl epsiv//sup 4/ and decoding complexity of order n//spl epsiv//sup 2/. The other construction is recursive. The encoding/decoding algorithms assume a delay of n bits i.e. rely on the entire codeword. The encoding/decoding complexity behaves roughly as n//spl epsiv//sup 2/ and n//spl epsiv/, respectively.

Minimal vectors in linear codes arise in numerous applications, particularly, in constructing decoding algorithms and studying linear secret sharing schemes. However, properties and structure of minimal vectors have been largely unknown. We prove basic properties of minimal vectors in general linear codes. Then we characterize minimal vectors of a given weight and compute their number in several classes of codes, including the Hamming codes and second-order Reed-Muller codes. Further, we extend the concept of minimal vectors to codes over rings and compute them for several examples. Turning to applications, we introduce a general gradient-like decoding algorithm of which minimal-vectors decoding is an example. The complexity of minimal-vectors decoding for long codes is determined by the size of the set of minimal vectors. Therefore, we compute this size for long randomly chosen codes. Another example of algorithms in this class is given by zero-neighbors decoding. We discuss relations between the two decoding methods. In particular, we show that for even codes the set of zero neighbors is strictly optimal in this class of algorithms. This also implies that general asymptotic improvements of the zero-neighbors algorithm in the frame of gradient-like approach are impossible. We also discuss a link to secret-sharing schemes.

Two deterministic algorithms of computing the weight spectra of binary cyclic codes are presented. These algorithms have the lowest known complexity for cyclic codes. For BCH codes of lengths 63 and 127, several first coefficients of the weight spectrum in number sufficient to evaluate the bounded distance decoding error probability are computed.>

The authors consider the construction of balanced error-correcting codes with distance close to half of the block length and bounded running digital sum. Use of these codes in cascade constructions allows derivation of a number of classes of DC-constrained codes of various lengths. The mathematical framework underlying the code construction is the theory of (incomplete) exponential sums. Essentially, the authors consider a class of codes formed by values of Legendre symbols of polynomials of bounded degree on the set of residues modulo a prime. In particular, taking linear polynomials, they obtain the Hadamard codes. Applying well-known estimates of the exponential sums, they compute the code parameters and prove that the proposed codes are in fact DC constrained.>