We present a modified version of the Bravyi-Terhal bound that applies to quantum codes defined by local parity-check constraints on aD-dimensional lattice quotient. Specifically, we consider a quotient ZD/Λ of ZDof cardinality ℓ, where Λ is someD-dimensional sublattice of ZD: we suppose that every vertex of this quotient indexesmqubits of a stabilizer codeC, which therefore has lengthn=mℓ. We prove that if all stabilizer generators act on qubits whose indices lie within a ball of radius ρ, then the minimum distancedof the code satisfiesd≤m√ γD( √D+ 4ρ)ℓD−1/D, where γDis theD-dimensional Hermite constant. We then apply this bound to derive an upper bound on the minimum distance of Abelian Two-Block Group Algebra (2BGA) codes whose parity-check matrices have the form [A|B] with each submatrix representing an element of a group algebra over a finite abelian group.

We provide a general framework for bounding the block error threshold of a linear code$C \subseteq \mathbb{F}_{2}^{N}$over the erasure channel in terms of its bit error threshold. Our approach relies on understanding the minimum support weight of any$r$-dimensional subcode of$C$, for all small values of$r$. As a proof of concept, we use our machinery to obtain a new proof of the celebrated result that Reed-Muller codes achieve capacity on the erasure channel with respect to block error probability.

We introduce and analyse an efficient decoder for quantum Tanner codes that can correct adversarial errors of linear weight. Previous decoders for quantum low-density parity-check codes could only handle adversarial errors of weight \(O(\sqrt{n\log n})\) . We also work on the link between quantum Tanner codes and the lifted product codes of Panteleev and Kalachev and show that our decoder can be adapted to the latter. The decoding algorithm alternates between sequential and parallel procedures and converges in linear time.

Constructing quantum low-density parity-check (LDPC) codes with a minimum distance that grows faster than a square root of the length has been a major challenge of the field. With this challenge in mind, we investigate constructions that come from high-dimensional expanders, in particular Ramanujan complexes. These naturally give rise to very unbalanced quantum error correcting codes that have a large $X$-distance but a much smaller $Z$-distance. However, together with a classical expander LDPC code and a tensoring method that generalizes a construction of Hastings and also the Tillich–Zémor construction of quantum codes, we obtain quantum LDPC codes whose minimum distance exceeds the square root of the code length and whose dimension comes close to a square root of the code length. When the ingredient is a 2-dimensional Ramanujan complex, or the 2-skeleton of a 3-dimensional Ramanujan complex, we obtain a quantum LDPC code of minimum distance $n^{1/2}\log^{1/2}n$. We then exploit the expansion properties of the complex to devise the first polynomial-time algorithm that decodes above the square root barrier for quantum LDPC codes. Using a 3-dimensional Ramanujan complex, we also obtain an overall quantum code of minimum distance $n^{1/2}\log n$, which sets a new record for quantum LDPC codes.

Kitaev’s toric code is arguably the most studied quantum code and is expected to be implemented in future generations of quantum computers. The renormalisation decoders introduced by Duclos-Cianci and Poulin exhibit one of the best trade-offs between accuracy and efficiency, with a time complexity inO(nlog2n). One question that was left open is how they handle worst-case or adversarial errors, i.e. what is the order of magnitude of the smallest weight of an error pattern that will be wrongly decoded. We initiate such a study involving a simple hard-decision and deterministic version of a renormalisation decoder. We exhibit an uncorrectable error pattern whose weight scales liked1/2and prove that the decoder corrects all error patterns of weight less than 5/6dlog2(6/5), wheredis the minimum distance of the toric code.

Kitaev's toric code is arguably the most studied quantum code and is expected to be implemented in future generations of quantum computers. The renormalisation decoders introduced by Duclos-Cianci and Poulin exhibit one of the best trade-offs between efficiency and speed, but one question that was left open is how they handle worst-case or adversarial errors, i.e. what is the order of magnitude of the smallest weight of an error pattern that will be wrongly decoded. We initiate such a study involving a simple hard-decision and deterministic version of a renormalisation decoder. We exhibit an uncorrectable error pattern whose weight scales like d1/2and prove that the decoder corrects all error patterns of weight less than $\frac{5}{6}{d^{{{\log }_2}(6/5)}}$, where d is the minimum distance of the toric code.

We introduce and analyse an efficient decoder for quantum Tanner codes that can correct adversarial errors of linear weight. Previous decoders for quantum low-density parity-check codes could only handle adversarial errors of weight . We also work on the link between quantum Tanner codes and the Lifted Product codes of Panteleev and Kalachev, and show that our decoder can be adapted to the latter. The decoding algorithm alternates between sequential and parallel procedures and converges in linear time.

We introduce sequential and parallel decoders for quantum Tanner codes. When the Tanner code construction is applied to a sufficiently expanding square complex with robust local codes, we obtain a family of asymptotically good quantum low-density parity-check codes. In this case, our decoders provably correct arbitrary errors of weight linear in the code length, respectively in linear or logarithmic time. The same decoders are easily adapted to the expander lifted product codes of Panteleev and Kalachev. Along the way, we exploit recently established bounds on the robustness of random tensor codes to give a tighter bound on the minimum distance of quantum Tanner codes.

Tanner codes are long error correcting codes obtained from short codes and a graph, with bits on the edges and parity-check constraints from the short codes enforced at the vertices of the graph. Combining good short codes together with a spectral expander graph yields the celebrated expander codes of Sipser and Spielman, which are asymptotically good classical LDPC codes. In this work we apply this prescription to the left-right Cayley complex that lies at the heart of the recent construction of a c3locally testable code by Dinur et at. Specifically, we view this complex as two graphs that share the same set of edges. By defining a Tanner code on each of those graphs we obtain two classical codes that together define a quantum code. This construction can be seen as a simplified variant of the Panteleev and Kalachev asymptotically good quantum LDPC code, with improved estimates for its minimum distance. This quantum code is closely related to the Dinur et at. code in more than one sense: indeed, we prove a theorem that simultaneously gives a linearly growing minimum distance for the quantum code and recovers the local testability of the Dinur et at. code.

In this paper we introduce Ouroboros, a new family of Key Exchange protocols based on coding theory. The protocols propose a middle ground between the cryptosystems based on$\mathsf {QC}$-$\mathsf {MDPC}$codes, which feature small parameter sizes, but have a security reduction to two problems: the syndrome decoding problem and the indistinguishability of the code, and the HQC protocol, which features bigger parameters but has a security reduction to the syndrome decoding problem only. Ouroboros features a reduction to the syndrome decoding problem with only a small overhead compared to the$\mathsf {QC}$-$\mathsf {MDPC}$based cryptosystems. The approach is based on an ideal structure and also works for the rank metric. This yields a simple, secure and efficient approach for key exchange, the Ouroboros family of protocols. For the Hamming metric we obtain the same type of parameters (and almost the same simple decoding) as for$\mathsf {MDPC}$based cryptosystems, but with a security reduction to decoding random quasi-cyclic codes in the Random Oracle Model. This represents a reduction of up to 38% on the public key size compared to HQC, for the most secure parameters. For the rank metric, we obtain better parameters than for RQC, saving up to 31% on the public key for the most secure set of parameters, using non homogeneous errors in Ouroboros. In this full version, the protocol and decoding algorithm have been slightly improved, additional details are given in the security proof, and the protocol is fully described for the rank metric.

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.

We consider oblivious transfer protocols performed over binary symmetric channels in a malicious setting where parties will actively cheat if they can. We provide constructions purely based on coding theory that achieve an explicit positive rate, the essential ingredient being the existence of linear codes whose Schur products are asymptotically good.

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.

Constructing quantum LDPC codes with a minimum distance that grows faster than a square root of the length has been a major challenge of the field. With this challenge in mind, we investigate constructions that come from high-dimensional expanders, in particular Ramanujan complexes. These naturally give rise to very unbalanced quantum error correcting codes that have a large X-distance but a much smaller Z-distance. However, together with a classical expander LDPC code and a tensoring method that generalises a construction of Hastings and also the Tillich-Zemor construction of quantum codes, we obtain quantum LDPC codes whose minimum distance exceeds the square root of the code length and whose dimension comes close to a square root of the code length. When the ingredient is a 3-dimensional Ramanujan complex, we show that its 2-systole behaves like a square of the log of the complex size, which results in an overall quantum code of minimum distance n1/2logn, and sets a new record for quantum LDPC codes. When we use a 2-dimensional Ramanujan complex, or the 2-skeleton of a 3-dimensional Ramanujan complex, we obtain a quantum LDPC code of minimum distance n1/2log1/2n. We then exploit the expansion properties of the complex to devise the first polynomial time algorithm that decodes above the square root barrier for quantum LDPC codes.

We give a linear-time erasure list-decoding algorithm for expander codes. More precisely, let r > 0 be any integer. Given an inner codeC0of length d, and a d-regular bipartite expander graph G with n vertices on each side, we give an algorithm to list-decode the codeC=C(G,C0) of length nd from approximately δδrnd erasures in time n·poly (d2r/δ), where δ and δrare the relative distance and the r'th generalized relative distance ofC0, respectively. To the best of our knowledge, this is the first linear-time algorithm that can list-decode expander codes from erasures beyond their (designed) distance of approximately δ2nd. To obtain our results, we show that an approach similar to that of (Hemenway and Wootters, Information and Computation, 2018) can be used to obtain such an erasure-list-decoding algorithm with an exponentially worse dependence of the running time on r and δ; then we show how to improve the dependence of the running time on these parameters.

In the model that has become known as “Perfectly Secure Message Transmission” (PSMT), a sender Alice is connected to a receiver Bob through n parallel two-way channels. A computationally unbounded adversary Eve controls t of these channels, meaning she can acquire and alter any data that is transmitted over these channels. The sender Alice wishes to communicate a secret message to Bob privately and reliably, i.e. in such a way that Eve gains no information about the message while Bob is able to recover it completely. We focus on PSMT protocols that work in two transmission rounds for n = 2t + 1. We break from previous work by following a conceptually simpler blueprint. This has two consequences: first, we obtain improved efficiency, namely, we reduce the previously best-known communication complexity, i.e. the number of transmitted bits necessary to communicate a 1-bit secret, from O(n3log n) to O(n2log n). Our solution also reaches optimal transmission rate for a secret of size O(n log n), thus answering the hitherto open question of attaining a threshold below O(n2log n) bits. Second, our construction can be adapted to more general scenarios relevant to Network Coding, where the adversary is given more power.

We introduce a new family of rank metric codes: Low Rank Parity Check codes (LRPC), for which we propose an efficient probabilistic decoding algorithm. This family of codes can be seen as the equivalent of classical LDPC codes for the rank metric. We then use these codes to design cryptosystems à la McEliece: more precisely we propose two schemes for key encapsulation mechanism (KEM) and public key encryption (PKE). Unlike rank metric codes used in previous encryption algorithms -notably Gabidulin codes - LRPC codes have a very weak algebraic structure. Our cryptosystems can be seen as an equivalent of the NTRU cryptosystem (and also to the more recent MDPC code-based cryptosystem) in a rank metric context, due to the similar form of the public keys. The present paper is an extended version of the article introducing LRPC codes, with important new contributions. We have improved the decoder thanks to a new approach which allows for decoding of errors of higher rank weight, namely up to$\frac {2}{3}(n-k)$when the previous decoding algorithm only decodes up to$\frac {n-k}{2}$errors. Our codes therefore outperform the classical Gabidulin code decoder which deals with weights up to$\frac {n-k}{2}$. This comes at the expense of probabilistic decoding, but the decoding error probability can be made arbitrarily small. The new approach can also be used to decrease the decoding error probability of previous schemes, which is especially useful for cryptography. Finally, we introduce ideal rank codes, which generalize double-circulant rank codes and allow us to avoid known structural attacks based on folding. To conclude, we propose different parameter sizes for our schemes and we obtain a public key of 3337 bits for key exchange and 5893 bits for public key encryption, both for 128 bits of security.

Constellation shaping is necessary to approach channel capacity for information rates above 1 bit/dim. Probabilistic shaping shows a small gap to capacity, however a complex distribution matcher is required to modify the source distribution. Spherical shaping of lattice constellations also reduces the gap to capacity, but practical Voronoi shaping is feasible in small dimensions only. In this paper, our codebook is a real geometrically non-uniform Gaussian-like constellation. We prove that this discrete codebook achieves channel capacity when the number of points goes to infinity. Then we build a special mapping to interface between non-binary low-density codes and the codebook, allowing the code alphabet size to be equal to the square root of the codebook size. Excellent performance is shown with fast-encoding and practical iterative probabilistic decoding, e.g. 0.7 dB gap to capacity at 6 bits/s/Hz with a code defined over the ring Z/8Z.

We propose a framework for constructing efficient code-based encryption schemes that do not hide any structure in their public matrix. The framework is in the spirit of the schemes first proposed by Alekhnovich in 2003 and based on the difficulty of decoding random linear codes from random errors of low weight. We depart somewhat from Alekhnovich's approach and propose an encryption scheme based on the difficulty of decoding random quasi-cyclic codes. We propose two new cryptosystems instantiated within our framework: the hamming quasi-cyclic cryptosystem (HQC), based on the hamming metric, and the rank quasi-cyclic cryptosystem (RQC), based on the rank metric. We give a security proof, which reduces the indistinguishability under chosen plaintext attack security of our systems to a decision version of the well-known problem of decoding random families of quasi-cyclic codes for the hamming and rank metrics (the respective QCSD and RQCSD problems). We also provide an analysis of the decryption failure probability of our scheme in the Hamming metric case: for the rank metric there is no decryption failure. Our schemes benefit from a very fast decryption algorithm together with small key sizes of only a few thousand bits. The cryptosystems are very efficient for low encryption rates and are very well suited to key exchange and authentication. Asymptotically, for λ the security parameter, the public key sizes are respectively in O(λ2) for HQC and in O(λ 4/3) for RQC. Practical parameter compares well to the systems based on ring-learning parity with noise or the recent moderate density parity check codes system.

This paper deals with Low-Density Construction-A (LDA) lattices, which are obtained via Construction A from non-binary low-density parity-check codes. More precisely, a proof is provided that Voronoi constellations of LDA lattices achieve capacity of the AWGN channel under lattice encoding and decoding for every signal-to-noise ratio greater than 1. This is obtained after showing the same result for more general Construction-A lattice constellations. The theoretical analysis is carried out in a way that allows to describe how the prime number underlying Construction A behaves as a function of the lattice dimension. Moreover, no dithering is required in the transmission scheme, simplifying some previous solutions of the problem. Remarkably, capacity is achievable with LDA lattice codes whose parity-check matrices have constant row and column Hamming weights. Some expansion properties of random bipartite graphs constitute an extremely important tool for dealing with sparse matrices and allow to find a lower bound for the minimum Euclidean distance of LDA lattices in our ensemble.

In the model of Secure Network Coding, a sender is connected to several receivers by a network, i.e. a directed graph with a single source node and several destination nodes, where each node can perform operations on the values received via the incoming edges and sends the results via the outbound edges. An active adversary controls some of the edges; this means that he can read every symbol transmitted over the edges under his control and replace them with symbols of his choice. The goal of Secure Network Coding is to design protocols that allow transmission of a secret message from the sender to all receivers in a private and reliable way. Classically, only one-way communication (from sender to receivers) has been studied; in this setting, security can be guaranteed as long as the number of edges controlled by the adversary is less than one third of the network connectivity. In this paper, we present a procedure where receivers are allowed to send feedback to the sender; with this feature, security is guaranteed against a stronger adversary: namely, the number of corrupted edges only needs to be smaller than one half of the connectivity. Furthermore, like previous state-of-the-art work on the single-round scenario, our scheme is universal, i.e. it does not require knowledge of the network code.

We give a randomized reduction for the Rank Syndrome Decoding problem and Rank Minimum Distance problem for rank codes over extension fields. Our results are based on embedding linear codes in the Hamming space into linear codes over an extension field equipped with the rank metric. We prove that if any of the previous problems for the rank metric is in ZPP = RP∩coRP, then we would have NP = ZPP. We also give complexity results for the respective rank metric approximation problems.

We present an efficient decoding algorithm for constant rate quantum hyper graph-product LDPC codes which provably corrects adversarial errors of weight proportional to the code minimum distance, or equivalently to the square-root of the block length. The algorithm runs in time linear in the number of qubits, which makes its performance the strongest to date for linear-time decoding of quantum codes. The algorithm relies on expanding properties, not of the quantum code's factor graph directly, but of the factor graph of the original classical code it is constructed from.

Given a linear code C, one can define the dth power of C as the span of all componentwise products of d elements of C. A power of C may quickly fill the whole space. Our purpose is to answer the following question: does the square of a code typically fill the whole space? We give a positive answer, for codes of dimension k and length roughly (1/2)k2or smaller. Moreover, the convergence speed is exponential if the difference k(k+1)/2-n is at least linear in k. The proof uses random coding and combinatorial arguments, together with algebraic tools involving the precise computation of the number of quadratic forms of a given rank, and the number of their zeros.

We characterize product-maximum distance separable (PMDS) pairs of linear codes, i.e., pairs of codes C and D whose product under coordinatewise multiplication has maximum possible minimum distance as a function of the code length and the dimensions dim C and dim D. We prove in particular, for C = D, that if the square of the code C has minimum distance at least 2, and (C, C) is a PMDS pair, then either C is a generalized Reed-Solomon code, or C is a direct sum of self-dual codes. In passing we establish coding-theory analogues of classical theorems of additive combinatorics.

In this paper, we revisit the combinatorial error model of Mazumdar et al. that models errors in high-density magnetic recording caused by lack of knowledge of grain boundaries in the recording medium. We present new upper bounds on the cardinality/rate of binary block codes that correct errors within this model. All our bounds, except for one, are obtained using combinatorial arguments based on hypergraph fractional coverings. The exception is a bound derived via an information-theoretic argument. Our bounds significantly improve upon existing bounds from the prior literature.

The current best asymptotic lower bound on the minimum distance of quantum LDPC codes with a fixed non-zero rate is logarithmic in the blocklength. We propose a construction of quantum LDPC codes with fixed non-zero rate and prove that the minimum distance grows proportionally to the square root of the blocklength.

In this paper, we re-visit the combinatorial error model of Mazumdar et al. [3] that models errors in high-density magnetic recording caused by lack of knowledge of grain boundaries in the recording medium. We present new upper bounds on the cardinality/rate of binary block codes that correct errors within this model.

We address the problem of transmission of information over the AWGN channel using lattices. In particular, we will deal with previously introduced LDA lattices which are obtained by Construction A from LDPC codes over the finite field Fp. We will show how to build a particular ensemble of LDA lattices related to bipartite graphs with good expansion properties. We investigate the quality of this family under lattice decoding and show that a random member in it can be reliably decoded for any value of the channel noise variance up to Poltyrev limit. Values of p and the parameters for which optimal performance is guaranteed under lattice decoding are in accordance with the optimal parameters found experimentally under iterative decoding.

We study a construction of quantum LDPC codes proposed by MacKay, Mitchison, and Shokrollahi. It is based on the Cayley graph of \BBF2ntogether with a set of generators regarded as the columns of the parity-check matrix of a classical code. We give a general lower bound on the minimum distance of the quantum code in O(dn2) where d is the minimum distance of the classical code. This bound is logarithmic in the blocklength 2nof the quantum code. When the classical code is the [n,1,n] repetition code, we are able to compute the exact parameters of the associated quantum code which are [[2n, 2[(n+1)/2], 2[(n-1)/2]]].

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 describe a new family of integer lattices built from construction A and non-binary LDPC codes. An iterative message-passing algorithm suitable for decoding in high dimensions is proposed. This family of lattices, referred to as LDA lattices, follows the recent transition of Euclidean codes from their classical theory to their modern approach as announced by the pioneering work of Loeliger (1997), Erez, Litsyn, and Zamir (2004-2005). Besides their excellent performance near the capacity limit, LDA lattice construction is conceptually simpler than previously proposed lattices based on multiple nested binary codes and LDA decoding is less complex than real-valued message passing.

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.

We study a construction of Quantum LDPC codes proposed by MacKay, Mitchison and Shokrollahi in the draft [6]. It is based on the Cayley graph of F2ntogether with a set of generators regarded as the columns of the parity-check matrix of a classical code. We give a general lower bound on the minimum distance of the quantum code in O(dn2) where d is the minimum distance of the classical code. When the classical code is the [n, 1, n] repetition code, we are able to compute the exact parameters of the associated quantum code which are [[2n-1, 2 n/2, 2 n/2-1]].

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).

We are interested in percolation for a family of self-dual tilings of the hyperbolic plane. We achieve an upper bound on the critical probability for these tilings by taking appropriate finite quotients and associating them with a family of quantum CSS codes. We then relate the probability of percolation to the probability of a decoding error for these codes on the quantum erasure channel.

We design powerful low-density parity-check (LDPC) codes with iterative decoding for the block-fading channel. We first study the case of maximum-likelihood decoding, and show that the design criterion is rather straightforward. Since optimal constructions for maximum-likelihood decoding do not perform well under iterative decoding, we introduce a new family of full-diversity LDPC codes that exhibit near-outage-limit performance under iterative decoding for all block-lengths. This family competes favorably with multiplexed parallel turbo codes for nonergodic channels.

The current best asymptotic lower bound on the minimum distance of quantum LDPC codes with fixed non-zero rate is logarithmic in the block length. We build quantum LDPC codes with fixed non-zero rate and prove that their minimum distance grows proportionally to the square root of the block length.

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.

A new graph-based construction of generalized low density codes (GLD-Tanner) with binary BCH constituents is described. The proposed family of GLD codes is optimal on block erasure channels and quasi-optimal on block fading channels. Optimality is considered in the outage probability sense. A classical GLD code for ergodic channels (e.g., the AWGN channel, the i.i.d. Rayleigh fading channel, and the i.i.d. binary erasure channel) is built by connecting bitnodes and subcode nodes via a unique random edge permutation. In the proposed construction of full-diversity GLD codes (referred to as root GLD), bitnodes are divided into 4 classes, subcodes are divided into 2 classes, and finally both sides of the Tanner graph are linked via 4 random edge permutations. The study focuses on non-ergodic channels with two states and can be easily extended to channels with 3 states or more.

We propose a method for extracting an errorless secret key in a continuous-variable quantum key distribution protocol, which is based on Gaussian modulation of coherent states and homodyne detection. The crucial feature is an eight-dimensional reconciliation method, relying on the algebraic properties of octonions. By using this coding scheme with an appropriate signal-to-noise ratio, the distance for secure continuous-variable quantum key distribution can be significantly extended.

We show how to build full-diversity product codes under both iterative encoding and decoding over non-ergodic channels, in presence of block erasure and block fading. The concept of a rootcheck or a root subcode is introduced by generalizing the same principle recently invented for low-density parity-check codes. We also describe some channel related graphical properties of the new family of product codes, a family referred to as root product codes.

Fuzzy commitment schemes, introduced as a link between biometrics and cryptography, are a way to handle biometric data matching as an error-correction issue. We focus here on finding the best error-correcting code with respect to a given database of biometric data. We propose a method that models discrepancies between biometric measurements as an erasure and error channel, and we estimate its capacity. We then show that two-dimensional iterative min-sum decoding of properly chosen product codes almost reaches the capacity of this channel. This leads to practical fuzzy commitment schemes that are close to theoretical limits. We test our techniques on public iris and fingerprint databases and validate our findings.

The Gilbert-Varshamov (GV) bound states that the maximum size A2(n, d) of a binary code of length n and minimum distance d satisfies A2(n, d)ges2n/V(n, d-1) where V(n, d)=Sigmai=0d(in) stands for the volume of a Hamming ball of radius d. Recently, Jiang and Vardy showed that for binary nonlinear codes this bound can be improved to A2(n, d)gescn2n/(V(n, d-1)) for c a constant and d/nges0.499. In this paper, we show that certain asymptotic families of linear binary [n, n/2] random double circulant codes satisfy the same improved GV bound.

The Gilbert-Varshamov bound states that the maximum size A2(n,d) of a binary code of length n and minimum distance d satisfies A2(n,d)ges2n/V(n,d-1) where V(n,d)=XiEdi=0(Eni) stands for the volume of a Hamming ball of radius d. Recently Jiang and Vardy showed that for binary non-linear codes this bound could be improved to A2(n,d)gescn2n/V(n,d -1) for c a constant and d/nles0.499. In this paper we show that certain asymptotic families of linear binary [n, n/2] double circulant codes satisfy the same improved Gilbert-Varshamov bound

We investigate the minimum distance of structured LDPC codes with two variable nodes of degree 2 per parity-check equation and show that their minimum distance is a sub-linear power function of the code length

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.

In this correspondence, we present an interleaving scheme that yields quasi-cyclic turbo codes. We prove that randomly chosen members of this family yield with probability almost 1 turbo codes with asymptotically optimum minimum distance, i.e., growing as a logarithm of the interleaver size. These interleavers are also very practical in terms of memory requirements and their decoding error probabilities for small block lengths compare favorably with previous interleaving schemes.

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

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.

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.

This paper presents the study of interleavers that both use a reduced amount of random choice and moderate algebraic structure. The interleaver is essentially chosen at random among a family that produces quasicyclic turbo codes. A typical interleaver produces a turbo code with minimum distance log N. For moderate lengths these interleavers turn out to be also quite practical, comparing favorably with S-random interleavers in a number of instances.

This paper discusses the generalised coset schemes for the wire-tap channel and its application to biometrics. The main contribution to this problem is threefold:1) information-theoretic approach and model the situation by involving wire-tap channel models, 2) generalisation of Wyner's coset coding scheme to the case when both the main channel and the wiretap channel are noisy, 3) complete solution to the original problem by making use of LDPC codes, with low decoding complexity. This paper also prove that this scheme i.e., the use of linear codes, achieve the Shannon-capacity of the system.

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.

The Gaussian isoperimetric inequality states that among all sets in /spl Ropf//sup n/ with prescribed Gaussian measure, the half-spaces have minimal Gaussian perimeter. We apply this result to Voronoi regions of codes in Euclidean space and obtain a surprisingly precise description of how the maximum-likelihood decoding error probability varies as a function of the minimum Euclidean distance.

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.

Fault diagnosis of multiprocessor systems motivates the following graph-theoretic definition. A subset C of points in an undirected graph G=(V, E) is called an identifying code if the sets B(v)/spl cap/C consisting of all elements of C within distance one from the vertex v are different. We also require that the sets B(v)/spl cap/C are all nonempty. We take G to be the infinite square lattice with diagonals and show that the density of the smallest identifying code is at least 2/9 and at most 4/17.

Sipser and Spielman (see ibid., vol.42, p.1717-22, Nov. 1996) have introduced a constructive family of asymptotically good linear error-correcting codes-expander codes-together with a simple parallel algorithm that will always remove a constant fraction of errors. We introduce a variation on their decoding algorithm that, with no extra cost in complexity, provably corrects up to 12 times more errors.

In an undirected graph G=(V,E), a subset $C \subseteq V$ is called an identifying code if the sets $B_1(v) \cap C$ consisting of all elements of C within distance one from the vertex v are nonempty and different. We take G to be the infinite hexagonal grid and show that the density of any identifying code is at least 16/39 and that there is an identifying code of density 3/7.

We build a class of pseudo-random error correcting codes, called generalized low density codes (GLD), from the intersection of two interleaved block codes. GLD code performance approaches the channel capacity limit and the GLD decoder is based on simple and fast SISO (soft input-soft output) decoders of smaller block codes. GLD codes are a special case of Tanner codes and a generalization of Gallager's LDPC codes. It is also proved by an ensemble performance argument that these codes are asymptotically good in the sense of the minimum distance criterion. The flexibility in selecting the parameters of GLD codes makes them suitable for small and large block length forward error correcting schemes.

We aim here to show how some known Ramanujan Cayley graphs yield error-correcting codes that are asymptotically optimal in the class of cycle codes of graphs. The main reason why known constructions of Ramanujan graphs yield good cycle codes is that the number of their cycles of a given length behaves essentially like that of random regular graphs. More precisely, we show that for actual constructions of Ramanujan graphs of degree $\Delta$ which are bipartite, and for the double cover of known Ramanujan graphs which are not bipartite,the number of cycles of length 2l is $\BigOe(\Delta-1+\varepsilon)^{2l}$ (for every $\varepsilon > 0$), which is about what one could expect from a random regular graph of degree $\Delta$. Furthermore, it is possible to show that this property guarantees the highest possible error probability p that the corresponding cycle codes can sustain, among the class of cycle codes of $\Delta$-regular graphs. This gives a constructive answer to an early problem in coding theory, namely, determining what is asymptotically the best possible performance of cycle codes of graphs when submitted to the binary symmetric channel.

We study partitions of the space $\mathbb{F}_2^n $ of all the binary n-tuples into disjoint sets, where each set is an additive cosec of a given set V. Such a partition is called a tiling of $\mathbb{F}_2^n $ and denoted $(V,A)$, where A is the set of cosec representatives. We give a sufficient condition for a set V to be a tile in terms of the cardinality of $V + V$. We then employ this condition to classify all tilings with sets of small cardinality. Further, periodicity of tilings in $\mathbb{F}_2^n $ is discussed, and a simple construction of nonperiodic tilings of $\mathbb{F}_2^n $ is presented for all $n \geq 6$. It is also shown that the nonperiodic tiling of $\mathbb{F}_2^6 $ is unique. A tiling $(V,A)$ is said to be proper if V generates $\mathbb{F}_2^n $; it is said to be full rank if both V and A generate $\mathbb{F}_2^n $. We show that, in general, the classification of tilings can be reduced to the study of proper tilings. We then prove that any tiling may be decomposed into smaller tilings that are either trivial or have full rank. Existence of full-rank tilings is exhibited by showing that each tiling is uniquely associated with a perfect binary code. Moreover, it is shown that periodic full-rank tilings may be further decomposed into smaller tilings, and then the existence of nonperiodic full-rank tilings is deduced. Finally, we generalize the well-known Lloyd theorem, originally stated for tilings by spheres, for the case of arbitrary tilings.

Given a subset X of vertices of the n-cube (i.e., the n-dimensional Hamming space), we are interested in the solution of the traveling salesman problem; namely, the minimal length of a cycle passing through all vertices of X. For a given number M, we estimate the maximum of these lengths when X ranges over all possible choices of sets of M vertices. Asymptotically, our estimates show that for a number M of vertices growing exponentially in n, the maximum is attained for a code with maximal possible minimum distance.

We study a wide class of problems in coding theory for which we consider two different formulations: in terms of incidence matrices and in terms of hypergraphs. These problems are dealt with using a greedy algorithm due to Stein (1974) and Lovasz (1975). Some examples, including constructing covering codes, codes for conflict resolution, separating systems, source encoding with distortion, etc., are given a unified treatment. Under certain conditions derandomization can be performed, leading to an essential reduction in the complexity of the constructions.

We define and estimate the threshold probability /spl theta/ of a linear code, using a theorem of Margulis (1974) originally conceived for the study of the probability of disconnecting a graph. We then apply this concept to the study of the erasure and Z-channels, for which we propose linear coding schemes that admit simple decoding. We show that /spl theta/ is particularly relevant to the erasure channel since linear codes achieve a vanishing error probability as long as p/spl lesspl theta/, where p is the probability of erasure. In effect, /spl theta/ can be thought of as a capacity notion designed for codes rather than for channels. Binomial codes haven the highest possible /spl theta/ (and achieve capacity). As for the Z-channel, a subcapacity is derived with respect to the linear coding scheme. For a transition probability in the range ]log (3/2); 1[, we show how to achieve this subcapacity. As a by-product we obtain improved constructions and existential results for intersecting codes (linear Sperner families) which are used in our coding schemes.>

We derive new asymptotic bounds for generalized distances. Our approach extends the classical Hamming, Plotkin, and Elias bounds. The latter bound involves extending the definition of generalized distances to nonlinear codes.>

A binary intersecting code is a linear code with the property that any two nonzero codewords have intersecting supports. These codes appear in a wide variety of contexts and applications, e.g., multiple access, cryptography, and information theory. This paper is devoted partly to the study of intersecting codes, and partly to their use in constructing large t-independent families of binary vectors. The latter subject has by now been extensively studied and has application in VLSI testing, defect correction, E-biased probability spaces, and derandomization. By concatenation methods we construct codes with the highest known fate asymptotically. We then generalize the concept to t-wise intersecting codes: we give bounds on the achievable rate of such codes, both existential and constructive. We show how t-wise intersecting codes can be used to obtain (t+1)-independent families. With this method we obtain improved asymptotical constructions of t-independent families. Complexity issues are discussed.>

A problem raised by R.L. Rivest and A. Shamir (1982), namely, constructing write-once-memory (WOM) codes capable of error correction, is considered. The authors call a (n,m,t)-WOM code a scheme that allows t successive writings of m arbitrary bits (i.e., one message among 2/sup m/) on a WOM of size n. WOM codes have been studied from an information-theoretic viewpoint by J.K. Wolf et al. (1984) and constructed using classical coding theory by G.D. Cohen et al. (1986, 1987) (for example, with parameters, (23,11,3), (2/sup m-1/,m,2/sup m-2/+2/sup m-4/+1)). The authors adapt those methods in order to solve the problem raised by Rivest. Large classes of easily decodable single-error-correcting WOM codes are obtained.>