In this paper, we consider the recently introduced concept of \emph{function-correcting codes (FCCs) with data protection}, which provide a certain level of error protection for the data and a higher level of protection for a desired function on the data. These codes are denoted by $(f\!:\!d_d,d_f)$-FCC, where $d_d$ is the minimum distance of the code and $d_f$ denotes the minimum distance between those codewords that correspond to different function values of a function $f:\mathbb{F}_q^k \to \mathrm{Im}(f)$, with $d_f \geq d_d$. We use a distance graph on a code based on the pairwise distances of its codewords, and show conditions under which a code cannot work as a \emph{strict} $(f\!:\!d_d,d_f)$-FCC, that is, code for which $d_f > d_d$. We then consider some well-known classes of codes, such as perfect codes and maximum distance separable (MDS) codes, and show that they cannot be used as \emph{strict} $(f\!:\!d_d,d_f)$-FCCs.

Function-correcting codes (FCCs) are designed to provide error protection for the value of a function computed on the data. Existing work typically focuses solely on protecting the function value and not the underlying data. In this work, we propose a general framework that offers protection for both the data and the function values. Since protecting the data inherently contributes to protecting the function value, we focus on scenarios where the function value requires stronger protection than the data itself. We first introduce a more general approach and a framework for function-correcting codes that incorporates data protection along with protection of function values. A two-step construction procedure for such codes is proposed, and bounds on the optimal redundancy of general FCCs with data protection are reported. Using these results, we exhibit examples that show that data protection can be added to existing FCCs without increasing redundancy. Using our two-step construction procedure, we present explicit constructions of FCCs with data protection for specific families of functions, such as locally bounded functions and the Hamming weight function. We associate a graph called minimum-distance graph to a code and use it to show that perfect codes and maximum distance separable (MDS) codes cannot provide additional protection to function values over and above the amount of protection for data for any function. Then we focus on linear FCCs and provide some results for linear functions, leveraging their inherent structural properties. To the best of our knowledge, this is the first instance of FCCs with a linear structure. Finally, we generalize the Plotkin and Hamming bounds well known in classical error-correcting coding theory to FCCs with data protection.

We introduce function-correcting partition codes (FCPCs), which are a natural generalization of function-correcting codes (FCCs). An FCPC is defined directly on a partition of the message space, rather than on a specific target function. We show that any FCC for a function $f$ is exactly an FCPC with respect to the domain partition induced by $f$, which makes these codes a natural generalization of FCCs. We use the join of domain partitions to construct a single code that protects multiple functions simultaneously. We define the notions of partition gains to measure the bandwidth saved by using a single FCPC for multiple functions instead of constructing separate FCCs for each function. We derive general lower and upper bounds on the redundancy of such FCPCs and illustrate the achievable gains through examples. We specialize this concept of using single code for protecting multiple functions to linear functions via coset partition of the intersection of their kernels. We also present explicit FCPC constructions for locally bounded partitions and grouped weight partitions. Then, we associate a partition graph with any given partition of $\mathbb{F}_q^k$, and show that the existence of a suitable clique in this graph yields a set of representative information vectors that achieves the optimal redundancy. Using the existence of a full-size clique in the weight partition and support partition, we obtain lower and upper bounds on the optimal redundancy of FCPCs for these partitions. We introduce the notion of a block-preserving contraction for a partition, which helps reduce the problem size of finding optimal redundancy for an FCPC. We further show that such a contraction exists for all weight-based partitions. Finally, we observe that FCPCs naturally provide a form of partial privacy in the sense that only the domain partition of the function needs to be revealed to the transmitter.

In this paper, we present secure distributed matrix multiplication (SDMM) schemes over the complex numbers with good numerical stability and small mutual information leakage by utilizing polynomial interpolation with roots of unity. Furthermore, we give constructions utilizing the real numbers by first encoding the real matrices to smaller complex matrices using a technique we callcomplexification. These schemes over the real numbers enjoy many of the benefits of the schemes over the complex numbers, including good numerical stability, but are computationally more efficient. To analyze the numerical stability and the mutual information leakage, we give some bounds on the condition numbers of Vandermonde matrices whose evaluation points are roots of unity.

Function-correcting codes (FCCs) are designed to provide error protection for the value of a function computed on the data. Existing work typically focuses solely on protecting the function value and not the underlying data. In this work, we propose a general framework that offers protection for both the data and the function values. Since protecting the data inherently contributes to protecting the function value, we focus on scenarios where the function value requires stronger protection than the data itself. A two-step construction procedure for such codes is proposed, and bounds on the optimal redundancy of general FCCs with data protection are reported. Using these results, we exhibit examples that show that data protection can be added to existing FCCs without increasing redundancy. Using our two-step construction procedure, we present explicit constructions of FCCs with data protection for specific families of functions, such as locally bounded functions and the Hamming weight function. We associate a graph calledminimum-distance graphto a code and use it to show that perfect codes and maximum distance separable (MDS) codes cannot provide additional protection to function values over and above the amount of protection for data for any function. Then we focus on linear FCCs and provide some results for linear functions, leveraging their inherent structural properties. While FCCs for linear functions have been considered earlier in the literature, to the best of our knowledge, the linearity of the FCC itself has not been studied before. Finally, we generalize the Plotkin and Hamming bounds well known in classical error-correcting coding theory to FCCs with data protection.

The connection between secret sharing and matroid theory is well established. In this paper, we generalize the concepts of secret sharing and matroid ports to q-polymatroids. Specifically, we introduce the notion of an access structure on a vector space, and consider properties related to duality, minors, and the relationship to q-polymatroids. Finally, we show how rank-metric codes give rise to secret sharing schemes within this framework.

In federated learning, multiple parties train models locally and share their parameters with a central server, which aggregates them to update a global model. To address the risk of exposing sensitive data through local models, secure aggregation via secure multiparty computation has been proposed to enhance privacy. At the same time, perfect privacy can only be achieved by a uniform distribution of the "masked" local models to be aggregated. This raises a problem when working with real-valued data, as there is no measure on the reals that is invariant under the masking operation, and hence information leakage is bound to occur. Shifting the data to a finite field circumvents this problem, but as a downside runs into an inherent accuracy–complexity tradeoff issue due to fixed-point modular arithmetic as opposed to floating-point numbers that can simultaneously handle numbers of varying magnitudes. In this paper, a novel secure parameter aggregation method is proposed that employs the torus rather than a finite field. This approach guarantees perfect privacy for each party’s data by utilizing the uniform distribution on the torus, while avoiding accuracy losses. Experimental results show that the new protocol performs similarly to the model without secure aggregation while maintaining perfect privacy. Compared to the finite field secure aggregation, the torus-based protocol can in some cases significantly outperform it in terms of model accuracy and cosine similarity, hence making it a safer choice.

In this paper, we introduce a class of functions that assume only a limited number λ of values within a given Hamming ρ-ball and call them locally (ρ,λ)-bounded functions. We develop function-correcting codes (FCCs) for a subclass of these functions and propose an upper bound on the redundancy of FCCs. The bound is based on the minimum length of an error-correcting code with a given number of codewords and a minimum distance. Furthermore, we provide a sufficient optimality condition for FCCs when λ = 4. We also demonstrate that any function can be represented as a locally (ρ,λ)-bounded function, illustrating this with a representation of Hamming weight distribution functions. Furthermore, we present another construction of function-correcting codes for Hamming weight distribution functions.

Abstract We prove the equivalence between the Ring Learning With Errors (RLWE) and the Polynomial Learning With Errors (PLWE) problems for the maximal totally real subfield of the $$2^r 3^s$$ 2 r 3 s th cyclotomic field for $$r \ge 3$$ r ≥ 3 and $$s \ge 1$$ s ≥ 1 . Moreover, we describe a fast algorithm for computing the product of two elements in the ring of integers of these subfields. This multiplication algorithm has quasilinear complexity in the dimension of the field, as it makes use of the fast Discrete Cosine Transform (DCT). Our approach assumes that the two input polynomials are given in a basis of Chebyshev-like polynomials, in contrast to the customary power basis. To validate this assumption, we prove that the change of basis from the power basis to the Chebyshev-like basis can be computed with $${\mathcal {O}}(n \log n)$$ O ( n log n ) arithmetic operations, where n is the problem dimension. Finally, we provide a heuristic and theoretical comparison of the vulnerability to some attacks for the pth cyclotomic field versus the maximal totally real subextension of the 4pth cyclotomic field for a reasonable set of parameters of cryptographic size.

Linear computations over quantum many-to-one communication networks offer opportunities for communication cost improvements through schemes that exploit quantum entanglement among transmitters to achieve superdense coding gains, combined with classical techniques such as interference alignment. The problem becomes much more broadly accessible if suitable abstractions can be found for the underlying quantum functionality via classical black box models. This work formalizes such an abstraction in the form of an “N-sum box”, a black box generalization of a two-sum protocol of Song et al. with recent applications to N-server private information retrieval. The N-sum box has a communication cost of N qudits and classical output of a vector of$N~q$-ary digits linearly dependent (via an$N \times 2N$transfer matrix) on$2N$classical inputs distributed among N transmitters. We characterize which transfer matrices are feasible by our construction, both with and without the possibility of additional locally invertible classical operations at the transmitters and receivers. Furthermore, we provide a sample application to Cross-Subspace Alignment (CSA) schemes to obtain efficient instances of Quantum Private Information Retrieval (QPIR) and Quantum Secure Distributed Batch Matrix Multiplication (QSDBMM). We first describe N-sum boxes based on maximal stabilizers and we then consider non-maximal-stabilizer-based constructions to obtain an instance of Quantum Symmetric Private Information Retrieval.

In this paper, we propose a novel construction for secure distributed matrix multiplication (SDMM) based on algebraic geometry (AG) codes, which we call the PoleGap SDMM scheme. The proposed construction is inspired by the Gap Additive Secure Polynomial (GASP) code, where so-called gaps in a certain polynomial are utilized to achieve higher communication rates. Our construction considers the gaps in a Weierstrass semigroup of a rational place in an algebraic function field to achieve a similar increase in the rate. This construction shows that there is potential in utilizing AG codes and their subcodes in SDMM since we demonstrate a better performance compared to state-of-the-art schemes in some parameter regimes.

A new framework for interference alignment in secure and private information retrieval (PIR) from colluding servers is proposed, generalizing the original cross-subspace alignment (CSA) codes proposed by Jia, Sun, and Jafar. The general scheme is built on algebraic geometry codes and explicit constructions with replicated storage are given over curves of genus zero and one. It is shown that the proposed scheme offers interesting tradeoffs between the field size, file size, number of colluding servers, and the total number of servers. When the field size is fixed, this translates in some cases to higher retrieval rates than those of the original scheme. In addition, the new schemes exist also in cases where the original ones do not.

Private information retrieval from a single server is considered, utilizing the hardness of the decoding problem of random linear codes. Presented is a modified version of the first code-based single-server computational PIR scheme proposed by Holzbaur, Hollanti, and Wachter-Zeh in [Holzbaur et al., “Computational Code-Based Single-Server Private Information Retrieval”, 2020 IEEE ISIT]. The original scheme was broken in [Bordage et al., “On the privacy of a code-based single-server computational PIR scheme”, Cryptogr. Comm., 2021] by an attack arising from highly probable rank differences in sub-matrices of the user's query. Here, this attack is now circumvented by ensuring that the sub-matrices have negligible rank difference. Furthermore, the rank difference cannot be attributed to the desired file index, thereby ensuring privacy. In the case of retrieving multiple files, the rate of the modified scheme is largely unaffected and at par with the original scheme.

In this paper, a general framework for linear secure distributed matrix multiplication (SDMM) is introduced. The model allows for a neat treatment of straggling and Byzantine servers via a star product interpretation as well as simplified security proofs. Known properties of star products also immediately yield a lower bound for the recovery threshold as well as an upper bound for the number of colluding workers the system can tolerate. Another bound on the recovery threshold is given by the decodability condition, which generalizes a bound for GASP codes. The framework produces many of the known SDMM schemes as special cases, thereby providing unification for the previous literature on the topic. Furthermore, error behavior specific to SDMM is discussed and interleaved codes are proposed as a suitable means for efficient error correction in the proposed model. Analysis of the error correction capability under natural assumptions about the error distribution is also provided, largely based on well-known results on interleaved codes. Error detection and other error distributions are also discussed.

Linear computations over quantum many-to-one communication networks offer opportunities for communication cost improvements through schemes that exploit quantum entanglement among transmitters to achieve superdense coding gains, combined with classical techniques such as interference alignment. The problem becomes much more broadly accessible if suitable abstractions can be found for the underlying quantum functionality via classical black box models. This work formalizes such an abstraction in the form of an “N-sum box”, a black box generalization of a two-sum protocol of Song et al. with recent applications to$N$-server private information retrieval. The N- sum box has a communication cost of$N$qudits and classical output of a vector of$N$q-ary digits linearly dependent (via an N x 2N transfer matrix) on 2N classical inputs distributed among$N$transmitters. We characterize which transfer matrices are feasible by our construction, both with and without the possibility of additional locally invertible classical operations at the transmitters and receivers.

The Gram matrix of a matrix A is defined as AAT(or ATA). Computing the Gram matrix is an important operation in many applications, such as linear regression with the least squares method, where the explicit solution formula includes the Gram matrix of the data matrix. Secure distributed matrix multiplication (SDMM) can be used to compute the product of two matrices using the help of worker servers. If a Gram matrix were computed using SDMM, the data matrix would need to be encoded twice, which causes an unnecessary overhead in the communication cost. We propose a new scheme for this purpose called secure distributed Gram matrix multiplication (SDGMM). It can leverage the advantages of computing a Gram matrix instead of a regular matrix product.

This work considers the problem of distributing matrix multiplication over the real or complex numbers to helper servers, such that the information leakage to these servers is close to being information-theoretically secure. These servers are assumed to be honest-but-curious, i.e., they work according to the protocol, but try to deduce information about the data. The problem of secure distributed matrix multiplication (SDMM) has been considered in the context of matrix multiplication over finite fields, which is not always feasible in real world applications. We present two schemes, which allow for variable degree of security based on the use case and allow for colluding and straggling servers. We analyze the security and the numerical accuracy of the schemes and observe a trade-off between accuracy and security.

This paper is eligible for the Jack Keil Wolf ISIT Student Paper Award. In this work, a flexible and robust private information retrieval (PIR) scheme based on binary non-maximum distance separable (non-MDS) codes is considered. This combines previous works on PIR schemes based on transitive non-MDS codes on one hand, and PIR from MDS-coded Byzantine and nonresponsive servers on the other hand. More specifically, a PIR scheme employing binary Reed–Muller (RM) codes tolerant to colluding, Byzantine, and non-responsive servers is constructed, and bounds for the achievable rates are derived under certain conditions. The construction of such schemes turns out to be much more involved than for MDS codes. Namely, the binary query vectors have to be selected with great care to hit the desired information sets, which is technically challenging as will be shown.

In this paper, a general framework for linear secure distributed matrix multiplication (SDMM) is introduced. The model allows for a neat treatment of straggling and Byzantine servers via a star product interpretation as well as simplified security proofs. Known properties of star products also immediately yield a lower bound for the recovery threshold, as well as an upper bound for the number of colluding workers the system can tolerate. It produces many of the known SDMM schemes as special cases, hence providing unification for the previous literature on the topic. Furthermore, error behavior specific to SDMM is discussed and interleaved codes are proposed as a suitable means for efficient error correction in the proposed model. Analysis of the error correction capability is also provided, largely based on well-known results on interleaved codes.

In quantum private information retrieval (QPIR), a user retrieves a classical file from multiple servers by downloading quantum systems without revealing the identity of the file. The QPIR capacity is the maximal achievable ratio of the retrieved file size to the total download size. In this paper, the capacity of QPIR from MDS-coded and colluding servers is studied for the first time. Two general classes of QPIR, called stabilizer QPIR and dimension-squared QPIR induced from classical strongly linear PIR are defined, and the related QPIR capacities are derived. For the non-colluding case, the general QPIR capacity is derived when the number of files goes to infinity. A general statement on the converse bound for QPIR with coded and colluding servers is derived showing that the capacities of stabilizer QPIR and dimension-squared QPIR induced from any class of PIR are upper bounded by twice the classical capacity of the respective PIR class. The proposed capacity-achieving scheme combines the star-product scheme by Freij-Hollantiet al.and the stabilizer QPIR scheme by Songet al.by employing (weakly) self-dual Reed–Solomon codes.

This work considers the problem of privately outsourcing the computation of a matrix product over a finite field${\mathbb {F}}_{q}$to$N$helper servers. These servers are considered to be honest but curious,i.e., they behave according to the protocol but will try to deduce information about the user’s data. Furthermore, any set of up to$X$servers is allowed to share their data. Previous works considered this collusion a hindrance and the download cost of the schemes increases with growing$X$. We propose to utilize such linkage between servers to the user’s advantage by allowing servers to cooperate in the computational task. This leads to a significant gain in the download cost for the proposed schemes. The gain naturally comes at the cost of increased communication load between the servers. Hence, the proposed cooperative schemes can be understood as outsourcing both computational cost and communication cost. Both information–theoretically secure and computationally secure schemes are considered, showing that allowing information leakage that is computationally hard to utilize will lead to further gains. The proposed server cooperation is then exemplified for specific secure distributed matrix multiplication (SDMM) schemes and linear private information retrieval (PIR). Similar ideas naturally apply to many other use cases as well, but not necessarily always with lowered costs.

For high-rate linear systematic maximum distance separable (MDS) codes, most early constructions could initially optimally repair all the systematic nodes but not all the parity nodes. Fortunately, this issue was first solved by Liet al.in (IEEE Trans. Inform. Theory, 64(9), 6257-6267, 2018), where a transformation that can convert any nonbinary MDS array code into another one with desired properties was proposed. However, the transformation does not work for binary MDS array codes. In this paper, we address this issue by proposing another generic transformation that can convert any$[n, k]$binary MDS array code into a new one, which endows any$r=n-k\ge 2$chosen nodes with optimal repair bandwidth and optimal rebuilding access properties, and at the same time, preserves the normalized repair bandwidth/rebuilding access for the remaining$k$nodes under some conditions. As two immediate applications, we show that 1) by applying the transformation multiple times, any binary MDS array code can be converted into one with optimal rebuilding access for all nodes, 2) any binary MDS array code with optimal repair bandwidth or optimal rebuilding access for the systematic nodes can be converted into one with the corresponding optimality property for all nodes.

In this paper, we study the problem ofprivate and secure distributed matrix multiplication (PSDMM), where a user having a private matrix$A$and$N$non-colluding servers sharing a library of$L$($L>1$) matrices$B^{(0)}, B^{(1)},\ldots,B^{(L-1)}$, for which the user wishes to compute$AB^{(\theta)}$for some$\theta \in [0, L$) without revealing any information of the matrix$A$to the servers, and keeping the index$\theta $private to the servers. Previous work is limited to the case that the shared library (i.e.,the matrices$B^{(0)}, B^{(1)},\ldots,B^{(L-1)}$) is stored across the servers in a replicated form and schemes are very scarce in the literature, there is still much room for improvement. In this paper, we propose two PSDMM schemes, where one is limited to the case that the shared library is stored across the servers in a replicated form but has a better performance than state-of-the-art schemes in that it can achieve a smaller recovery threshold and download cost. The other one focuses on the case that the shared library is stored across the servers in an MDS-coded form, which requires less storage in the servers. The second PSDMM code does not subsume the first one even if the underlying MDS code is degraded to a repetition code as they are totally two different schemes.

In this work, two practical concepts related to private information retrieval (PIR) are introduced and coinedfull support-rankPIR andstrongly linearPIR. Being of full support-rank is a technical, yet natural condition required to prove a converse result for a capacity expression and satisfied by almost all currently known capacity-achieving schemes, while strong linearity is a practical requirement enabling implementation over small finite fields with low subpacketization degree. Then, the capacity of MDS-coded, linear, full support-rank PIR in the presence of colluding servers is derived, as well as the capacity of symmetric, linear PIR with colluding, adversarial, and nonresponsive servers for the recently introduced concept of matched randomness. This positively settles the capacity conjectures stated by Freij-Hollantiet al.and Tajeddineet al.in the presented cases. It is also shown that, further restricting to strongly-linear PIR schemes with deterministic linear interference cancellation, the so-called star product scheme proposed by Freij-Hollantiet al.is essentially optimal and induces no capacity loss.

In the classical private information retrieval (PIR) setup, a user wants to retrieve a file from a database or a distributed storage system (DSS) without revealing the file identity to the servers holding the data. In the quantum PIR (QPIR) setting, a user privately retrieves a classical file by receiving quantum information from the servers. The QPIR problem has been treated by Song et al. in the case of replicated servers, both with and without collusion. QPIR over [n, k] maximum distance separable (MDS) coded servers was recently considered by Allaix et al., but the collusion was essentially restricted to t = n -$k$servers in the sense that a smaller$t$would not improve the retrieval rate. In this paper, the QPIR setting is extended to allow for retrieval with high rate for any number of colluding servers$t$with 1 ≤$t$≤$n$- k. Similarly to the previous cases, the rates achieved are better than those known or conjectured in the classical counterparts, as well as those of the previously proposed coded and colluding QPIR schemes. This is enabled by considering the stabilizer formalism and weakly self-dual generalized Reed-Solomon (GRS) star product codes.

Consider the problem of a user having a private matrix$A$and$N$non-colluding servers sharing a library of$L (L > \ 1)$matrices$B^{(0)}, B^{(1)},\ldots, B^{(L-1)}$, for which the user wishes to compute$AB^{(\theta)}$for some$\theta\in[0, L)$without revealing any information of the matrix$A$to the servers, and keeping the index$\theta$private to the servers. This problem is known as private and secure distributed matrix multiplication (PSDMM) and is supposed to have wide application potential. However, studies of PSDMM are still scarce in the literature. In this paper, we propose a new efficient private and secure distributed matrix multiplication coding scheme, which has a better performance than state-of-the-art schemes in that it achieves a smaller recovery threshold and download cost as well as providing a more flexible tradeoff between the upload and download costs.

Abstract This paper presents a new alphabet-dependent bound for codes with hierarchical locality. Then, the complete list of possible localities is derived for a class of codes obtained by deleting specific columns from a Simplex code. This list is used to show that these codes are optimal codes with hierarchical locality.

A private information retrieval (PIR) scheme allows a user to retrieve a file from a database without revealing any information on the file being requested. As of now, PIR schemes have been proposed for several kinds of storage systems, including replicated and MDS-coded systems. However, the problem of constructing PIR schemes on regenerating codes has been sparsely considered. A regenerating code is a storage code whose codewords are distributed among nodes, enabling efficient storage of files, as well as low-bandwidth retrieval of files and repair of nodes. Minimum-bandwidth regenerating (MBR) codes define a family of regenerating codes allowing a node repair with optimal bandwidth. Rashmi, Shah, and Kumar obtained a large family of MBR codes using the product-matrix (PM) construction. In this work, a new PIR scheme over PM-MBR codes is designed. The inherent redundancy of the PM structure is used to reduce the download communication complexity of the scheme. A lower bound on the PIR capacity of MBR-coded PIR schemes is derived, showing an interesting storage space vs. PIR rate trade-off compared to existing PIR schemes with the same reconstruction capability. The present scheme also outperforms a recent PM-MBR PIR construction of Dorkson and Ng.

The design of lattice coset codes for wiretap channels is considered. Bounds on the eavesdropper's correct decoding probability and information leakage are first revisited. From these bounds, it is explicit that both the information leakage and error probability are controlled by the average flatness factor of the eavesdropper's lattice, which we further interpret geometrically. It is concluded that the minimization of the (average) flatness factor of the eavesdropper's lattice leads to the study of well-rounded lattices, which are shown to be among the optimal in order to achieve these minima. Constructions of some well-rounded lattices are also provided.

In the classical private information retrieval (PIR) setup, a user wants to retrieve a file from a database or a distributed storage system (DSS) without revealing the file identity to the servers holding the data. In the quantum PIR (QPIR) setting, a user privately retrieves a classical file by downloading quantum systems from the servers. The QPIR problem has been treated by Song et al. in the case of replicated servers, both without collusion and with all but one servers colluding. In this paper, the QPIR setting is extended to account for maximum distance separable (MDS) coded servers. The proposed protocol works for any [n, k]-MDS code and t-collusion with t = n - k. Similarly to the previous cases, the rates achieved are better than those known or conjectured in the classical counterparts.

A new computational private information retrieval (PIR) scheme based on random linear codes is presented. A matrix of messages from a McEliece scheme is used to query the server with carefully chosen errors. The server responds with the sum of the scalar multiple of the rows of the query matrix and the files. The user recovers the desired file by erasure decoding the response. Contrary to code-based cryptographic systems, the scheme presented here enables to use truly random codes, not only codes disguised as such. Further, we show the relation to the so-called error subspace search problem and quotient error search problem, which we assume to be difficult, and show that the scheme is secure against attacks based on solving these problems.

We define and analyze low-rank parity-check (LRPC) codes over extension rings of the finite chain ring Zpr, where p is a prime and r is a positive integer. LRPC codes have originally been proposed by Gaborit et al. (2013) over finite fields for cryptographic applications. The adaption to finite rings is inspired by a recent paper by Kamche et al. (2019), which constructed Gabidulin codes over finite principle ideal rings with applications to space-time codes and network coding. We give a decoding algorithm based on simple linear-algebraic operations. Further, we derive an upper bound on the failure probability of the decoder. The upper bound is valid for errors whose rank is equal to the free rank.

Private information retrieval (PIR) is the problem of privately retrieving one out of M original files from N severs, i.e., each individual server gains no information on the identity of the file that the user is requesting. Usually, the M files are replicated or encoded by a maximum distance separable (MDS) code and then stored across the N servers. Compared to mere replication, MDS-coded servers can significantly reduce the storage overhead. Particularly, PIR from minimum storage regenerating (MSR) coded servers can simultaneously reduce the repair bandwidth when repairing failed servers. Existing PIR protocols from MSR-coded servers either require large sub-packetization levels or are not capacity-achieving. In this paper, a PIR protocol from MDS array codes is proposed, subsuming PIR from MSR-coded servers as a special case. Particularly, only the case of non-colluding, honest-but-curious servers is considered. The retrieval rate of the new PIR protocol achieves the capacity of PIR from MDS-/MSR-coded servers. By choosing different MDS array codes, the new PIR protocol can have varying advantages when compared with existing protocols, e.g., 1) small sub-packetization, 2) (near-)optimal repair bandwidth, 3) implementable over the binary field F2.

In this paper, the problem of providing privacy to users requesting data over a network from a distributed storage system (DSS) is considered. The DSS, which is considered as the multi-terminal destination of the network from the user's perspective, is encoded by a maximum rank distance (MRD) code to store the data on these multiple servers. A private information retrieval (PIR) scheme ensures that a user can request a file without revealing any information on which file is being requested to any of the servers. In this paper, a novel PIR scheme is proposed, allowing the user to recover a file from a storage system with low communication cost, while allowing some servers in the system to collude in the quest of revealing the identity of the requested file. The network is modeled as a random linear network, i.e., all nodes of the network forward random (unknown) linear combinations of incoming packets. Both error-free and erroneous random linear networks are considered.

In this work, we first prove the capacity of coded, linear symmetric private information retrieval (SPIR) in the presence of colluding, adversarial, and nonresponsive servers, giving a positive closure to the conjecture stated by Tajeddine et al. It is also shown that, further restricting to strongly-linear PIR schemes with linear interference cancellation, the so-called star product scheme proposed by Freij-Hollanti et al. is optimal. This observation enables to prove the capacity of strongly-linear (non-symmetric) PIR schemes for any number of files. Further, it also provides a positive proof in this practical special case for the conjectures stated in the asymptotic regime by Freij-Hollanti et al. and Tajeddine et al.

In a user-private information retrieval (UPIR) scheme, a set of users collaborate to retrieve files from a database without revealing to observers which participant in the scheme requested the file. To achieve privacy, users retrieve files from the database in response to anonymous requests posted to message spaces; assuming that each message space can be accessed by a subset of the participants in the scheme. Privacy with respect to the database is easily achieved, but privacy with respect to coalitions of other users within the scheme is sensitive to the choice of incidence structure determining which users can access each message space. Earlier schemes were based on pairwise balanced designs and symmetric designs, and involved at most one step of message passing to retrieve a file. We propose a new class of UPIR schemes based on generalised quadrangles (GQs), which need up to two steps of message passing in each file retrieval. We introduce a new message passing protocol in which messages are encrypted. Even using this protocol, previously proposed schemes are compromised by finite coalitions of users. We construct a family of GQ-UPIR schemes which maintain privacy with high probability even when $$O(n^{1/2-\epsilon })$$ users collude, where n is the total number of users in the scheme. We also show that a UPIR scheme based on any family of generalised quadrangles is secure against coalitions of $$O(n^{1/4-\epsilon })$$ users.

Private information retrieval (PIR) schemes for coded storage with colluding servers are presented, which are not restricted to maximum distance separable (MDS) codes. PIR schemes for general linear codes are constructed, and the resulting PIR rate is calculated explicitly. It is shown that codes with transitive automorphism groups yield the highest possible rates obtainable with the proposed scheme. In the special case of no server collusion, this rate coincides with the known asymptotic PIR capacity for MDS-coded storage systems. While many PIR schemes in the literature require field sizes that grow with the number of servers and files in the system, we focus especially on the case of a binary base field, for which Reed-Muller codes serve as an important and explicit class of examples.

Locally repairable codes (LRCs) have gained significant interest for the design of large distributed storage systems as they allow a small number of erased nodes to be recovered by accessing only a few others. Several works have thus been carried out to understand the optimal rate-distance tradeoff, but only recently the size of the alphabet has been taken into account. In this paper, a novel definition of locality is proposed to keep track of the precise number of nodes required for a local repair when the repair sets do not yield MDS codes. Then, a new alphabet-dependent bound is derived, which applies both to the new definition and the initial definition of locality. The new bound is based on consecutive residual codes and intrinsically uses the Griesmer bound. A special case of the bound yields both the extension of the Cadambe-Mazumdar bound and the Singleton-type bound for codes with locality $(r, {\delta})$, implying that the new bound is at least as good as these bounds. Furthermore, an upper bound on the asymptotic rate-distance tradeoff of LRCs is derived, and yields the tightest known upper bound for large relative minimum distances. Achievability results are also provided by deriving the locality of the family of Simplex codes together with a few examples of optimal codes.

The problem of private information retrieval (PIR) from coded storage systems with colluding, Byzantine, and unresponsive servers is considered. An explicit scheme using an [n, k] Reed-Solomon storage code is designed, protecting against t-collusion, and handling up to b Byzantine and r unresponsive servers, when n > k + t + 2b + r - 1. This scheme achieves a PIR rate of ((n - r - (k + 2b + t - 1))/n - r). In the case where the capacity is known, namely, when k = 1, it is asymptotically capacity achieving as the number of files grows. Finally, the scheme is adapted to symmetric PIR.

We consider a geographically constrained caching community where popular data files are cached on mobile terminals and distributed through Device-to-Device (D2D) communications. To ensure availability, data files are protected against user mobility, or churn, with select caching and erasure coding methods. Communication and storage costs are considered, with an objective of minimizing the consumption of radio resources, given an available storage size. We focus on finding the coding method that minimizes the overall cost. Closed-form expressions for the expected consumption of radio resources incurred by data delivery and redundancy maintenance are derived. Closed form transmission costs in a circular caching community with a specific node density and caching method are calculated, when cost obeys a power law of distance. Our results are illustrated by numerical examples and verified by extensive computer simulations.

A private information retrieval (PIR) scheme on coded storage systems with colluding, byzantine, and non-responsive servers is presented. Furthermore, the scheme can also be used for symmetric PIR in the same setting. An explicit scheme using an [n, k] generalized Reed-Solomon storage code is designed, protecting against t-collusion and handling up to b byzantine and r non-responsive servers, when n ≥ n1'=(ν+1)k+t+2b+r-1, for some integer ν ≥ 1. This scheme achieves a PIR rate of 1-[(k+2b+t+r-1)/(n'-r)]. In the case where the capacity is known, namely when k=1, it is asymptotically capacity achieving as the number of files grows.

Recently, information-theoretic private information retrieval (PIR) from coded storage systems has gained a lot of attention, and a general star product PIR scheme was proposed. In this paper, the star product scheme is adopted, with appropriate modifications, to the case of private (e.g., video) streaming. It is assumed that the files to be streamed are stored on n servers in a coded form, and the streaming is carried out via a convolutional code. The star product scheme is defined for this special case, and various properties are analyzed for two channel models related to straggling and Byzantine servers, both in the baseline case as well as with colluding servers. The achieved PIR rates for the given models are derived and, for the cases where the capacity is known, the first model is shown to be asymptotically optimal, when the number of stripes in a file is large. The second scheme introduced in this work is shown to be the equivalent of block convolutional codes in the PIR setting. For the Byzantine server model, it is shown to outperform the trivial scheme of downloading stripes of the desired file separately without memory.

The density of a code is the fraction of the coding space covered by packing balls centered around the codewords. A high density indicates that a code performs well when used as a uniform point-wise discretization of an ambient space. This paper investigates the density of codes in the complex Stiefel and Grassmann manifolds equipped with the chordal distance arising from an Euclidean embedding, including the unitary group as a special case. The choice of distance enables the treatment of the manifolds as subspaces of Euclidean hyperspheres. In this geometry, the densest packings are not necessarily equivalent to maximum-minimum-distance codes. Computing a code's density follows from computing: 1) the normalized volume of a metric ball and 2) the kissing radius, the radius of the largest balls one can pack around the codewords without overlapping. First, the normalized volume of a metric ball is evaluated by asymptotic approximations. The volume of a small ball can be well-approximated by the volume of a locally equivalent tangential ball. In order to properly normalize this approximation, the precise volumes of the manifolds induced by their spherical embedding are computed. For larger balls, a hyperspherical cap approximation is used, which is justified by a volume comparison theorem showing that the normalized volume of a ball in the Stiefel or Grassmann manifold is asymptotically equal to the normalized volume of a ball in its embedding sphere as the dimension grows to infinity. Then, bounds on the kissing radius are derived alongside corresponding bounds on the density. Unlike spherical codes or codes in flat spaces, the kissing radius of Grassmann or Stiefel codes cannot be exactly determined from its minimum distance. It is nonetheless possible to derive bounds on density as functions of the minimum distance. Stiefel and Grassmann codes have larger density than their image spherical codes when dimensions tend to infinity. Finally, the bounds on density lead to refinements of the standard Hamming bounds for Stiefel and Grassmann codes.

In this paper, we study lattice coding for Rician fading wireless channels. This is motivated in particular by preliminary studies suggesting the Rician fading model for millimeter-wavelength wireless communications. We restrict to lattice codes arising from rotations of Zn, and to a single-input single-output (SISO) channel. We observe that several lattice design criteria suggest the optimality of Hadamard rotations. For instance, we prove that Hadamard rotations maximize the diamond-packing density among all rotated Znlattices. Finally, we provide simulations to show that Hadamard rotations outperform optimal algebraic rotations and cross-packing lattices in the Rician channel.

In Private Information Retrieval (PIR), one wants to download a file from a database without revealing to the database which file is being downloaded. Much attention has been paid to the case of the database being encoded across several servers, subsets of which can collude to attempt to deduce the requested file. With the goal of studying the achievable PIR rates in realistic scenarios, we generalize results for coded data from the case of all subsets of servers of size t colluding, to arbitrary subsets of the servers. We investigate the effectiveness of previous strategies in this new scenario, and present new results in the case where the servers are partitioned into disjoint colluding groups.

Typically, locally repairable codes (LRCs) and regenerating codes have been studied independently of each other, and it has not been clear how the parameters of one relate to those of the other. In this paper, a novel connection between locally repairable codes and exact regenerating codes is established. Via this connection, locally repairable codes are interpreted as exact regenerating codes. Further, some of these codes are shown to perform better than time-sharing codes between minimum bandwidth regenerating and minimum storage regenerating codes.

Locally repairable codes (LRCs) are error correcting codes used in distributed data storage. Besides a global level, they enable errors to be corrected locally, reducing the need for communication between storage nodes. There is a close connection between almost affine LRCs and matroid theory which can be utilized to construct good LRCs and derive bounds on their performance. A generalized Singleton bound for linear LRCs with parameters (n; k; d; r; δ) was given in [N. Prakash et al., “Optimal Linear Codes with a Local-Error-Correction Property”, IEEE Int. Symp. Inf. Theory]. In this paper, a LRC achieving this bound is called perfect. Results on the existence and nonexistence of linear perfect (n; k; d; r; δ)-LRCs were given in [W. Song et al., “Optimal locally repairable codes”, IEEE J. Sel. Areas Comm.]. Using matroid theory, these existence and nonexistence results were later strengthened in [T. Westerbäck et al., “On the Combinatorics of Locally Repairable Codes”, Arxiv: 1501.00153], which also provided a general lower bound on the maximal achievable minimum distance dmax(n; k; r; δ) that a linear LRC with parameters (n; k; r; δ) can have. This article expands the class of parameters (n; k; d; r; δ) for which there exist perfect linear LRCs and improves the lower bound for dmax(n; k; r; δ). Further, this bound is proved to be optimal for the class of matroids that is used to derive the existence bounds of linear LRCs.

For many wiretap channel models asymptotically optimal coding schemes are known, but less effort has been put into actual realizations of wiretap codes for practical parameters. Bounds on the mutual information and error probability when using coset coding on a Rayleigh fading channel were recently established by Oggier and Belfiore, and the results in this paper build on their work. However, instead of using their ultimate inverse norm sum approximation, a more precise expression for the eavesdropper's probability of correct decision is used in order to determine a general class of good coset codes. The code constructions are based on well-rounded lattices arising from simple geometric criteria. In addition to new coset codes and simulation results, novel number-theoretic results on well-rounded ideal lattices are presented.

In this paper, locally repairable codes with all-symbol locality are studied. Methods to modify already existing codes are presented. It is also shown that, with high probability, a random matrix with a few extra columns guaranteeing the locality property is a generator matrix for a locally repairable code with a good minimum distance. The proof of the result provides a constructive method to find locally repairable codes. Finally, constructions of three infinite classes of optimal vector-linear locally repairable codes over a small alphabet independent of the code size are given.

This paper provides a link between matroid theory and locally repairable codes (LRCs) that are either linear or more generally almost affine. Using this link, new results on both LRCs and matroid theory are derived. The parameters (n, k, d, r, δ) of LRCs are generalized to matroids, and the matroid analog of the generalized singleton bound by Gopalan et al. for linear LRCs is given for matroids. It is shown that the given bound is not tight for certain classes of parameters, implying a nonexistence result for the corresponding locally repairable almost affine codes that are coined perfect in this paper. Constructions of classes of matroids with a large span of the parameters (n, k, d, r, δ) and the corresponding local repair sets are given. Using these matroid constructions, new LRCs are constructed with prescribed parameters. The existence results on linear LRCs and the nonexistence results on almost affine LRCs given in this paper strengthen the nonexistence and existence results on perfect linear LRCs given by Song et al.

In this article, the first general constructions of fast-decodable, more specifically (conditionally) g-group decodable, space-time block codes for the nonorthogonal amplify and forward (NAF) multiple-input multiple-output (MIMO) relay channel under the half-duplex constraint are proposed. In this scenario, the source and the intermediate relays used for data amplification are allowed to employ multiple antennas for data transmission and reception. The worst-case decoding complexity of the obtained codes is reduced by up to 75%. In addition to being fast-decodable, the proposed codes achieve full-diversity and have nonvanishing determinants, which has been shown to be useful for achieving the optimal diversity-multiplexing tradeoff (DMT) of the NAF channel. Furthermore, it is shown that the same techniques as in the cooperative scenario can be utilized to achieve fast-decodability for K-user MIMO multiple-access channel (MAC) space-time block codes. The resulting codes in addition exhibit the conditional nonvanishing determinant property which, for its part, has been shown to be useful for achieving the optimal MAC-DMT.

To optimize rotated multidimensional constellations over a single-input single-output Rayleigh fading channel, a family of rotation matrices is constructed for all dimensions which are a power of 2. This family is a one-parameter subgroup of the group of rotation matrices, and is located using a gradient descent scheme on this Lie group. The parameter defining the family is chosen to optimize the cutoff rate of the constellation. The optimal rotation parameter is computed explicitly for low signal-to-noise ratios. These rotations outperform full-diversity algebraic rotations in terms of cutoff rate at low signal-to-noise ratio (SNR) and bit error rate at high SNR in dimension n = 4. However, a quadrature amplitude modulation (QAM) constellation rotated by such a matrix lacks full diversity, in contrast with the conventional wisdom that good signal sets exhibit full diversity. A new notion of diversity, referred to as local diversity, is introduced to attempt to account for this behavior. Roughly, a locally fully diverse constellation is fully diverse only in small neighborhoods. A local variant of the minimum product distance is also introduced and is shown experimentally to be a superior predictor of constellation performance than the minimum product distance in dimension n = 4.

We consider lattice coset-coded transmissions over a wiretap channel with additive white Gaussian noise (AWGN). Examining a function that can be interpreted as either the legitimate receiver's error probability or the eavesdropper's correct decision probability, we rigorously show that, albeit offering simple bit labeling, orthogonal nested lattices are suboptimal for coset coding in terms of both the legitimate receiver's and the eavesdropper's probabilities.

We use a numerical algorithm on the Lie group of rotation matrices to obtain rotated constellations for Rayleigh fading channels. Our approach minimizes the union bound for the pairwise error probability to produce rotations optimized for a given signal-to-noise ratio. This approach circumvents explicit parametrization of rotation matrices, which has previously prevented robust numerical methods from being applied to constellation rotation. Our algorithm is applicable to arbitrary finite constellations in arbitrary dimensions, and one can thus apply our method to non-uniform constellations, which are of interest for practical concerns due to their ability to increase bit-interleaved coded modulation (BICM) capacity. We show how our rotations can improve the codeword error performance of non-uniform constellations, and we also apply our method to reproduce and improve rotations given by ideal lattices in cyclotomic fields.

With the goal of optimizing the CM (coded modulation) capacity of a finite constellation over a Rayleigh fading channel, we use one-parameter subgroups of the Lie group of rotation matrices to construct families of rotation matrices which optimize a certain objective function controlling the CM capacity. Our construction does not depend on any assumptions about the constellation or signal-to-noise ratio. We confirm the benefits of our construction for uniform and non-uniform constellations at a large range of SNR values through numerous simulations. We show that in two and four dimensions one can obtain a further potential increase in CM capacity by jointly considering non-uniform and rotated constellations.

In this paper we provide a link between matroid theory and locally repairable codes (LRCs) that are almost affine. The parameters (n, k, d, r) of LRCs are generalized to matroids. A bound on the parameters (n, k, d, r), similar to the bound in [P. Gopalan et al., “On the locality of codeword symbols,” IEEE Trans. Inf. Theory] for linear LRCs, is given for matroids. We prove that the given bound is not tight for a certain class of parameters, which implies a non-existence result for a certain class of optimal locally repairable almost affine codes. Constructions of optimal LRCs over small finite fields were stated as an open problem in [I. Tamo et al., “Optimal locally repairable codes and connections to matroid theory”, 2013 IEEE ISIT]. In this paper optimal LRCs which do not require a large field are constructed for certain classes of parameters.

The capacity of heterogeneous distributed storage systems under repair dynamics is studied. Examples of these systems include peer-to-peer storage clouds, wireless, and Internet caching systems. Nodes in a heterogeneous system can have different storage capacities and different repair bandwidths. Lower and upper bounds on the system capacity are given. These bounds depend on either the average resources per node, or on a detailed knowledge of the node characteristics. Moreover, the case in which nodes may be compromised by an eavesdropper is addressed and bounds on the secrecy capacity of the system are derived. One implication of these new results is that symmetric repair maximizes the capacity of a homogeneous system, which justifies the model widely used in the literature.

In this paper, we establish probability bounds for the correct decision of the eavesdropper over a MIMO Wiretap Channel, when coding using cyclic division algebras is used. We focus in particular on codebooks constructed from natural orders in Q-central quaternion algebras, which allows the resulting expressions to take a more explicit form.

The capacity of heterogeneous distributed storage systems under repair dynamics is studied. Examples of these systems include peer-to-peer storage clouds, wireless, and Internet caching systems. Nodes in a heterogeneous system can have different storage capacities and different repair bandwidths. Lower and upper bounds on the system capacity are given. These bounds depend on either the average resources per node, or on a detailed knowledge of the node characteristics. Moreover, the case in which nodes may be compromised by an adversary (passive or active) is addressed and bounds on the secure capacity of the system are derived. One implication of these new results is that symmetric repair maximizes the capacity of a homogeneous system, which justifies the model widely used in the literature.

In this paper, fast-decodable lattice code constructions are designed for the nonorthogonal amplify-and-forward (NAF) multiple-input multiple-output (MIMO) channel. The constructions are based on different types of algebraic structures, e.g. quaternion division algebras. When satisfying certain properties, these algebras provide us with codes whose structure naturally reduces the decoding complexity. The complexity can be further reduced by shortening the block length, i.e., by considering rectangular codes called less than minimum delay (LMD) codes.

Multiple-input double-output (MIDO) codes are important in the near-future wireless communications, where the portable end-user device is physically small and will typically contain at most two receive antennas. Especially tempting is the 4$\,\times\,$2 channel due to its immediate applicability in the digital video broadcasting (DVB). Such channels optimally employ rate-two space-time (ST) codes consisting of$(4\times 4)$matrices. Unfortunately, such codes are in general very complex to decode, hence setting forth a call for constructions with reduced complexity. Recently, some reduced complexity constructions have been proposed, but they have mainly been based on different ad hoc methods and have resulted in isolated examples rather than in a more general class of codes. In this paper, it will be shown that a family of division algebra based MIDO codes will always result in at least 37.5% worst-case complexity reduction, while maintaining full diversity and, for the first time, the nonvanishing determinant (NVD) property. The reduction follows from the fact that, similarly to the Alamouti code, the codes will be subsets of matrix rings of the Hamiltonian quaternions, hence allowing simplified decoding. At the moment, such reductions are among the best known for rate-two MIDO codes,. Several explicit constructions are presented and shown to have excellent performance through computer simulations.

Recently, extensive effort has been taken to build fast-decodable (FD) space-time (ST) codes, especially called for due to the limited power resources available for mobile receivers. Although there have been numerous successful attempts that have resulted in reduced decoding complexity for a certain number of transmit antennas, a general method for constructing FD codes for a wider range of antenna combinations and rates is still missing. Here, this problem is solved by introducing a totally general framework for constructing full-diversity FD codes with non-vanishing determinants (NVD).

In this paper, the probability of Eve the Eavesdropper's correct decision is considered both in the Gaussian and Rayleigh fading wiretap channels when using lattice codes for the transmission. First, it is proved that the secrecy function determining Eve's performance attains its maximum at y = 1 on all known extremal even unimodular lattices. This is a special case of a conjecture by Belfiore and Solé. Further, a very simple method to verify or disprove the conjecture on any given unimodular lattice is given. Second, preliminary analysis on the behavior of Eve's probability of correct decision in the fast fading wiretap channel is provided. More specifically, we compute the truncated inverse norm power sum factors in Eve's probability expression. The analysis reveals a performance-secrecy-complexity tradeoff: relaxing on the legitimate user's performance can significantly increase the security of transmission. The confusion experienced by the eavesdropper may be further increased by using skewed lattices, but at the cost of increased complexity.

Recently, several papers have been concentrating on reducing the decoding complexity of high-rate space-time codes. While the research has led to some impressive reductions in decoding complexity, the geometric methods so far used appear to have faced some fundamental limits. In this paper, we study what happens if we let go of the assumption of full diversity and study the possibility of reducing the complexity, while holding on to a high code rate, by reducing the length of codes. We will develop some tools that can be used to measure the changes we will encounter when reducing the code length. We will also study the achievable diversity-multiplexing gain trade-off (DMT) of codes with less than minimum delay (LMD) and discuss some code constructions.

Explicit code constructions for multiple-input multiple-output (MIMO) multiple-access channels (MAC) with$K$users are presented in this paper. The first construction is dedicated to the case of symmetric MIMO-MAC where all the users have the same number of transmit antennas$n_{t}$and transmit at the same level of per-user multiplexing gain$r$. Furthermore, we assume that the users transmit in an independent fashion and do not cooperate. The construction is systematic for any values of$K$,$n_{t}$and$r$. It is proved that this newly proposed construction achieves the optimal MIMO-MAC diversity-multiplexing gain tradeoff (DMT) provided by Tseat high-$\hbox{SNR}$regime.

The goal of this paper is to design fast-decodable space-time codes for four transmit and two receive antennas. The previous attempts to build such codes have resulted in codes that are not full rank and hence cannot provide full diversity or high coding gains. Extensive work carried out on division algebras indicates that in order to get, not only non-zero but perhaps even non-vanishing determinants (NVD) one should look at division algebras and their orders. To further aid the decoding, we will build our codes so that they consist of four generalized Alamouti blocks which allows decoding with reduced complexity. As far as we know, the resulting codes are the first having both reduced decoding complexity, and at the same time allowing one to give a proof of the NVD property.

Multiple-input double-output (MIDO) codes are important in future wireless communications, where the portable end-user device is physically small and will typically contain maximum two receive antennas. Especially tempting is the 4×2 channel, where the four transmitters can either be all at one station, or separated between two different stations. Such channels optimally employ rate-two space-time (ST) codes consisting of 4×4 matrices. Unfortunately, such codes are in general very complex to decode, the worst-case complexity being as high as N8, where N is the size of the complex signaling alphabet. Hence, constructions with reduced complexity are called for. One option, of course, is to use the rate-one codes such as the quasi-orthogonal codes. However, if full multiplexing, i.e., transmission of two symbols per channel use is to be maintained, this option has to be put aside. Recently, some reduced complexity constructions have been proposed, but they have mainly been based on ad hoc methods and have resulted in a specific code instead of a more general class of codes. In this paper, it will be shown that cyclic division algebra (CDA) based codes satisfying certain criteria will always result in at least 25% worst-case complexity reduction, while maintaining full diversity and even the non-vanishing determinant (NVD) property. The reduction follows from the fact that the codes will consist of four Alamouti blocks allowing simplified decoding. At the moment, such reduction is the best known for rate-two MIDO codes,. The code proposed in was the first one to provably fulfill the related algebraic properties, and shall be repeated here as an example. Further, a new low-complexity design resulting from the proposed criteria is presented, and shown to have excellent performance through simulations.

This paper considers certain aspects of some well-known multiple-input single-output (MISO) codes. In the first section it is proved how in some special cases the n + 1 MISO channel can be seen as consisting of several parallel MISO channels having less transmit antennas. It is also pointed out that unitary conjugation does not change the diversity-multiplexing tradeoff (DMT) of a code. These simple results are then applied to analyze the DMT of several well-known MISO codes. In particular its is proved that all the considered codes are DMT optimal. As a by-product of this study it is seen that the full-diversity quasi-orthogonal codes by Su and Xia are unitarily equivalent to division algebraic constructions. This relation is then used to place the constructions by Su and Xia into a wider context. In the latter part of the paper the 2 + 1 slow fading MISO channel is considered and it is proven that one of the previously proposed MISO multi-block codes (MB-codes) has a linear worst-case sphere decoding complexity.

In multiple-input-multiple-output (MIMO) communications, the notion of asymmetric channel refers to the situation when the number of transmit antennas is strictly larger than the number of receive antennas. Such channels can often be found in MIMO downlink transmissions. While existing cyclic-division-algebra (CDA)-based codes can still be employed to achieve the optimal diversity-multiplexing tradeoff (DMT) at high signal-to-noise ratio (SNR) regime, such codes cannot be directly decoded using, for example, the pure sphere decoding method. Although other means of decoding methods such as minimum mean square error generalized decision feedback equalizer (MMSE-GDFE) with lattice search and regularized lattice decoding are available, an alternative approach is to constrain the number of active transmit antennas in each channel use to be no larger than the number of receive antennas. The resulting system is coined constrained asymmetric MIMO system. Two general types of asymmetrical channels are considered in this paper, namely, 1) when there are two receive antennas and the number of transmit antennas is arbitrary, and 2) when the number of transmit antennas is one larger than the number of receive antennas. Explicit optimal transmission schemes as well as the corresponding code constructions for such constrained asymmetric MIMO channels are presented, and are shown to achieve the same DMT performance as their unconstrained counterparts.

An algebraic tool from the theory of central simple algebras is proposed to obtain families of complex matrices satisfying the conditional non-vanishing determinant (CNVD) property. Such property is of great use in e.g. the design of multiuser space-time (ST) codes, in which context it is not always crucial for the transmission matrix to be invertible. On the other hand, whenever it is invertible, it is important that it has a non-vanishing determinant. Also any submatrix of any subset of users multiplied with its transpose conjugate should preferably have a non-vanishing determinant, provided it is non-zero. In recent submissions by Lu et al. it has been shown that, with suitable multiplexing, such property yields a construction of space-time codes that achieve the optimal diversity-multiplexing tradeoff (DMT) of the multiple-input multiple-output (MIMO) multiple access channel (MAC) and outperform the previously known ST codes.

An explicit, systematic code construction for the symmetric MIMO (multi-input multi-output) multiple access (MAC) channel with any number of users and any numbers of transmit and receive antennas is presented in this paper. The users are assumed to transmit at the same level of multiplexing gain. This newly constructed code is proved to achieve the optimal MIMO-MAC diversity-multiplexing tradeoff.

In this paper, the need for the construction of asymmetric and multiblock space-time codes is discussed. Above the trivial puncturing method, i.e., switching off the extra layers in the symmetric multiple-input multiple-output (MIMO) setting, two more sophisticated asymmetric construction methods are proposed. The first method, called the block diagonal method (BDM), can be converted to produce multiblock space-time codes that achieve the diversity-multiplexing tradeoff (DMT). It is also shown that maximizing the density of the newly proposed block diagonal asymmetric space-time (AST) codes is equivalent to minimizing the discriminant of a certain order, a result that also holds as such for the multiblock codes. An implicit lower bound for the density is provided and made explicit for an important special case that contains e.g., the systems equipped with4Tx+2Rxantennas. Further, an explicit scheme achieving the bound is given. Another method proposed here is the smart puncturing method (SPM) that generalizes the subfield construction method proposed in earlier work by Hollanti and Ranto and applies to any number of transmitting and lesser receiving antennas. The use of the general methods is demonstrated by building explicit, sphere decodable codes using different cyclic division algebras (CDAs). Computer simulations verify that the newly proposed methods can compete with the trivial puncturing method, and in some cases clearly outperform it. The conquering construction exploiting maximal orders improves upon the punctured perfect code and the DjABBA code as well as the Icosian code. Also extensive DMT analysis is provided.

It is shown why the discriminant of a maximal order within a cyclic division algebra must be minimized in order to get the densest possible matrix lattices with a prescribed nonvanishing minimum determinant. Using results from class field theory, a lower bound to the minimum discriminant of a maximal order with a given center and index (= the number of Tx/Rx antennas) is derived. Also numerous examples of division algebras achieving the bound are given. For example, a matrix lattice with quadrature amplitude modulation (QAM) coefficients that has 2.5 times as many codewords as the celebrated Golden code of the same minimum determinant is constructed. Also, a general algorithm due to Ivanyos and Ronyai for finding maximal orders within a cyclic division algebra is described and enhancements to this algorithm are discussed. Also some general methods for finding cyclic division algebras of a prescribed index achieving the lower bound are proposed.

A method for constructing asymmetric space-time block codes (ASTBC) is proposed. This Smart Puncturing Method (SPM) generalizes the so-called subfield construction method (SCM) introduced in earlier work and applies to any antenna combination with #Rx≪#Tx as opposed to SCM, where the requirement is #Tx= m#Rx for some integer m. It has been shown that e.g. for 4Tx+2Rx antennas, the SCM performs equally or even better than the trivial puncturing method (TPM), but admits at the same time lower peak-to-mean power ratio. This is due to the fact that there are no zero slots in the code matrix but the information symbols are evenly spread into the matrix slots. The generalized method proposed in this paper is also based on cyclic division algebras (CDAs) and allows us to do the same for any number of receiving antennas #Rx≪#Tx.

In this paper, a systematic construction of asymmetric space-time codes that is a promising solution to the asymmetric coding problem is presented. By the asymmetric coding problem we mean that the number of receive antennas is strictly less than the number of transmit antennas in a MIMO communication system, and the task is to design a coding scheme that has an efficient decoding using e.g. a sphere decoder. Specifically, given any desired antenna selection pattern, the proposed construction will yield codes that are transmitted using exactly the specified pattern, and that can be easily decoded using sphere decoding or MMSE techniques. Our construction can be applied to any kinds of antenna selection patterns, including the one that equally-likely uses all possible n-subsets of the transmit antennas for some n. Moreover, the resulting codes are proved to achieve the diversity-multiplexing tradeoff associated with the designated selection pattern.

Recently, a family of full-rate, full-diversity space-time block codes (STBCs) for 2 times 2 multiple-input multiple-output (MIMO) channels was proposed in the works of Tirkkonen et al., using a combination of Clifford algebra and Alamouti structures, namely twisted space-time transmit diversity code. This family was recently rediscovered by Paredes et al., and they pointed out that such STBCs enable reduced-complexity maximum-likelihood (ML) decoding. Independently, the same STBCs were found in the work of Samuel and Fitz (2007) and named multi-strata space-time codes. In this paper we show how this code can be constructed algebraically from a particular cyclic division algebra (CDA). This formulation enables to prove that the code has the non-vanishing determinant (NVD) property and hence achieves the diversity-multiplexing tradeoff (DMT) optimality. The fact that the normalized minimum determinant is 1/radic(7) places this code in the second position with respect to the golden code, which exhibits a minimum determinant of 1/radic(5), and motivates the name silver code.

In this paper, we construct explicit rate-one, full-diversity, geometrically dense matrix lattices with large, nonvanishing determinants (NVDs) for four transmit antenna multiple-input–single-output (MISO) space-time (ST) applications. The constructions are based on the theory of rings of algebraic integers and related subrings of the Hamiltonian quaternions and can be extended to a larger number of Tx antennas. The usage of ideals guarantees an NVD larger than one and an easy way to present the exact proofs for the minimum determinants. The idea of finding denser sublattices within a given division algebra is then generalized to a multiple-input–multiple-output (MIMO) case with an arbitrary number of Tx antennas by using the theory of cyclic division algebras (CDAs) and maximal orders. It is also shown that the explicit constructions in this paper all have a simple decoding method based on sphere decoding. Related to the decoding complexity, the notion of sensitivity is introduced, and experimental evidence indicating a connection between sensitivity, decoding complexity, and performance is provided. Simulations in a quasi-static Rayleigh fading channel show that our dense quaternionic constructions outperform both the earlier rectangular lattices and the rotated quasi-orthogonal ABBA lattice as well as the diagonal algebraic space-time (DAST) lattice. We also show that our quaternionic lattice is better than the DAST lattice in terms of the diversity-multiplexing gain tradeoff (DMT).

In this paper, the need for the construction of multiple input-double output (MIDO) space-time block codes (STBCs) is discussed, concentrating on the case of four transmitters for simplicity. Above the trivial method, i.e. switching off the extra layers in the usual multiple input-multiple output (MIMO) setting, two smarter yet simple MIDO construction methods are proposed. The use of these general methods is then demonstrated by building explicit, sphere decodable codes using two different cyclic division algebras (CDAs). We verify by computer simulations that the newly proposed methods perform extremely well as opposed to the trivial construction.

In this paper, the need for the construction of asymmetric space-time block codes (ASTBCs) is discussed, mostly concentrating on the case of four transmitting and two receiving antennas for simplicity. Above the trivial puncturing method, i.e. switching off the extra layers in the symmetric multiple input-multiple output (MIMO) setting, a more sophisticated yet simple asymmetric construction method is proposed. This method can be converted to produce multi-block space-time codes that achieve the diversity-multiplexing (D-M) tradeoff. It is also shown that maximizing the density of the newly proposed codes is equivalent to minimizing the discriminant of a certain order. The use of the general method is then demonstrated by building explicit, sphere decodable codes using different cyclic division algebras (CDAs). We verify by computer simulations that the newly proposed method can compete with the puncturing method, and in some cases outperforms it. Our conquering construction exploiting maximal orders improves even upon the punctured perfect code and the DjABBA code.

We show why the discriminant of a maximal order within a cyclic division algebra must be minimized in order to get the densest possible matrix lattices with a prescribed non-vanishing minimal determinant. Using results from class field theory we derive a lower bound to the minimum discriminant of a maximal order with a given center and index (= the number of Tx/Rx antennas). We also give examples of division algebras achieving our bound. For example, we construct a matrix lattice with QAM coefficients that has (inside 'large' subsets of the signal space) 2.5 times as many codewords as the celebrated Golden code of the same minimum determinant. We also give another matrix lattice with coefficients from the hexagonal lattice with an even higher density

A means to construct dense, full-diversity STBCs from maximal orders in central simple algebras is introduced for the first time. As an example we construct an efficient ST lattice code with non-vanishing determinant for 4 transmit antenna MISO application. Also a general algorithm for testing the maximality of a given order is presented. By using a maximal order instead of just the ring of algebraic integers, the size of the code increases without losses in the minimum determinant. The usage of a proper ideal of a maximal order further improves the code, as the minimum determinant increases. Simulations in a quasi-static Rayleigh fading channel show that our lattice outperforms the DAST-lattice due to the properties described above.

We construct some geometrically dense matrix lattices with good minimum determinants for 4 transmit antenna MISO applications. The construction is based on the theory of rings of algebraic integers and related subrings of the Hamiltonian quaternions. Simulations in a quasi-static Rayleigh fading channel show that our dense quaternionic constructions outperform the earlier rectangular lattices as well as the DAST-lattice

Rate one full-diversity orthogonal designs for four transmit antennas are known not to exist so either rate, orthogonality or diversity is compromised. Use of algebraic number theory has lead to full-diversity, rate one code constructions. The use of regular representations of certain rings of algebraic integers and their Hamiltonian quaternionic counterparts to construct lattices of rank 8 that yield rate one full-diversity lattice constellation codes for four transmitting antennas is presented in this paper. The resulting codes require less transmission power per bit. Using number theoretic tools a general lower bound to the minimum Euclidean distance within the received constellation is computed