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Constrained codes are used to prevent errors from occurring in various data storage and data transmission systems. They can help in increasing the storage density of magnetic storage devices, in managing the lifetime of solid-state storage devices, and in increasing the reliability of data transmission over wires. Over the years, designing practical (complexity-wise) capacity-achieving constrained codes has been an area of research gaining significant interest. We recently designed various constrained codes based on lexicographic indexing. We introduced binary symmetric lexicographically-ordered constrained (S-LOCO) codes, $q$ -ary asymmetric LOCO (QA-LOCO) codes, and a class of two-dimensional LOCO (TD-LOCO) codes. These families of codes achieve capacity with simple encoding and decoding, and they are easy to reconfigure. We demonstrated that these codes can contribute to notable density and lifetime gains in magnetic recording (MR) and Flash systems, and they find application in other systems too. In this paper, we generalize our work on LOCO codes by presenting a systematic method that guides the code designer to build any constrained code based on lexicographic indexing once the finite set of data patterns to forbid is known. In particular, we connect the set of forbidden patterns directly to the cardinality of the LOCO code and most importantly to the rule that uncovers the index associated with a LOCO codeword. By doing that, we reveal the secret arithmetic of patterns, and make the design of such constrained codes significantly easier. We give examples illustrating the method via codes based on lexicographic indexing from the literature. We then design optimal (rate-wise) constrained codes for the new two-dimensional magnetic recording (TDMR) technology. Over a practical TDMR model, we show notable performance gains as a result of solely applying the new codes. Moreover, we show how near-optimal constrained codes for TDMR can be designed and used to further reduce complexity and error propagation. All the newly introduced LOCO codes are designed using the proposed general method, and they inherit all the desirable properties in our previously designed LOCO codes. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Previous works on age of information and erasure channels have dealt with specific models and computed the average age or average peak age for certain settings. In this paper, given a source that produces a letter every $T_{s}$ seconds and an erasure channel that can be used every $T_{c}$ seconds, we ask what is the coding strategy that minimizes the time-average “age of information” that an observer of the channel output incurs. We first analyze the case where the source alphabet and the channel-input alphabet have the same size. We show that a trivial coding strategy is optimal and a closed form expression for the age can be derived. We then analyze the case where the alphabets have different sizes. We use a random coding argument to bound the average age and show that the average age achieved using random codes converges to the optimal average age of linear block codes as the source alphabet becomes large. [ABSTRACT FROM AUTHOR]

Maximum distance separable (MDS) codes are very important in both theory and practice. There is a classical construction of a family of $[{2^{m}+1, 2u-1, 2^{m}-2u+3}]$ MDS codes for $1 \leq u \leq 2^{m-1}$ , which are cyclic, reversible and BCH codes over ${\mathrm {GF}}(2^{m})$. The objective of this paper is to study the quaternary subfield subcodes and quaternary subfield codes of a subfamily of the MDS codes for even $m$. A family of quaternary cyclic codes is obtained. These quaternary codes are distance-optimal in some cases and very good in general. Furthermore, two infinite families of 3-designs from these quaternary codes and their duals are presented. [ABSTRACT FROM AUTHOR]

Non-overlapping codes are a set of codewords in $\bigcup _{n \ge 2} \mathbb {Z}_{q}^{n}$ , where $\mathbb {Z}_{q} = \{0,1, {\dots },q-1\}$ , such that the prefix of each codeword is not a suffix of any codeword in the set, including itself; and for variable-length codes, a codeword does not contain any other codeword as a subword. In this paper, we investigate a generic method to generalize binary codes to $q$ -ary ones for $q > 2$ , and analyze this generalization on the two constructions given by Levenshtein (also by Gilbert; Chee, Kiah, Purkayastha, and Wang) and Bilotta, respectively. The generalization on the former construction gives large non-expandable fixed-length non-overlapping codes whose size can be explicitly determined; the generalization on the latter construction is the first attempt to generate $q$ -ary variable-length non-overlapping codes. More importantly, this generic method allows us to utilize the generating function approach to analyze the cardinality of the underlying $q$ -ary non-overlapping codes. The generating function approach not only enables us to derive new results, e.g., recurrence relations on their cardinalities, new combinatorial interpretations for the constructions, and the superior limit of their cardinalities for some special cases, but also greatly simplifies the arguments for these results. Furthermore, we give an exact formula for the number of fixed-length words that do not contain the codewords in a variable-length non-overlapping code as subwords. This thereby solves an open problem by Bilotta and induces a recursive upper bound on the maximum size of variable-length non-overlapping codes. [ABSTRACT FROM AUTHOR]

In this paper we prove several coordinate-ordering-free upper bounds on the insdel distances of linear codes. Our bounds are stronger than some previous known bounds. We apply these upper bounds to AGFC codes from some cyclic codes and one algebraic-geometric code with any rearrangement of coordinate positions. A strong upper bound on the insdel distances of Reed-Muller codes with the special coordinate ordering is also given. [ABSTRACT FROM AUTHOR]

Maximum Distance Separable (MDS) codes with a sparse and balanced generator matrix are appealing in distributed storage systems for balancing and minimizing the computational load. Such codes have been constructed via Reed-Solomon codes over large fields. In this paper, we focus on small fields. We prove that there exists an $[n,k]_{q}$ MDS code that has a sparse and balanced generator matrix for any $q\geq n-1$ provided that $n\leq 2k$ , by designing several algorithms with complexity running in polynomial time in $k$ and $n$. For the case $n>2k$ , we give some constructions for $q=n=p^{s}$ and $k=p^{e}m$ based on sumsets, when $e\leq s-2$ and $m\leq p-1$ , or $e=s-1$ and $m < \frac {p}{2}$. [ABSTRACT FROM AUTHOR]

In this paper, we construct four families of linear codes over finite fields from the complements of either the union of subfields or the union of cosets of a subfield, which can produce infinite families of optimal linear codes, including infinite families of (near) Griesmer codes. We also characterize the optimality of these four families of linear codes with an explicit computable criterion using the Griesmer bound and obtain many distance-optimal linear codes. In addition, by a more in-depth discussion on some special cases of these four families of linear codes, we obtain several classes of (distance-)optimal linear codes with few weights and completely determine their weight distributions. It is shown that most of our linear codes are self-orthogonal or minimal which are useful in applications. [ABSTRACT FROM AUTHOR]

In this paper, we present two new families of APN functions. The first family is in bivariate form $\big (x^{3}+xy^{2}+ y^{3}+xy, x^{5}+x^{4}y+y^{5}+xy+x^{2}y^{2} \big)\,\,\vphantom {_{\int _{\int }}}$ over ${\mathbb F}_{2^{m}}^{2}$. It is obtained by adding certain terms of the form $\sum _{i}(a_{i}x^{2^{i}}y^{2^{i}},b_{i}x^{2^{i}}y^{2^{i}})$ to a family of APN functions recently proposed by Gölo&gcaron;lu. The $\vphantom {_{\int _{\int }}}$ second family has the form $L(z)^{2^{m}+1}+vz^{2^{m}+1}$ over ${\mathbb F}_{{2^{3m}}}$ , which generalizes a family of APN functions by Bracken et al. from 2011. By calculating the $\Gamma $ -rank of the constructed APN functions over ${\mathbb F}_{2^{8}}$ and ${\mathbb F}_{2^{9}}$ , we demonstrate that the two families are CCZ-inequivalent to all known families. In addition, the two new families cover two known sporadic APN instances over ${\mathbb F}_{2^{8}}$ and ${\mathbb F}_{2^{9}}$ , which were found by Edel and Pott in 2009 and by Beierle and Leander in 2021, respectively. [ABSTRACT FROM AUTHOR]

One of the important unsolved problems in information theory is the conjecture that the undirected multiple-unicast network information capacity is the same as the routing capacity. This conjecture is verified only for a handful of networks and network classes. Moreover, only two explicit upper bounds on information capacity are known for general undirected networks: the sparsest cut bound and the linear programming bound. In this paper, we present an information-theoretic upper bound, called the partition bound, on the capacity of general undirected multiple-unicast networks. We show that a decision version problem of computing the bound is NP-complete. We present two classes of undirected multiple-unicast networks such that the partition bound is achievable by routing. Thus, the conjecture is proved for these classes of networks. Recently, the conjecture was proved for a new class of networks defined by properties relating to cut-set and source-sink paths. We show the existence of a network outside of this new class of networks such that the partition bound is achievable by routing. [ABSTRACT FROM AUTHOR]

Combinatorial designs are closely related to linear codes. Recently, some near MDS codes were employed to construct $t$ -designs by Ding and Tang, which settles the question as to whether there exists an infinite family of near MDS codes holding an infinite family of $t$ -designs for $t \geq 2$. This paper is devoted to the construction of infinite families of 3-designs and 2-designs from special equations over finite fields. First, we present an infinite family of almost MDS codes over ${\mathrm{ GF}}(p^{m})$ holding an infinite family of 3-designs. We then provide an infinite family of almost MDS codes over ${\mathrm{ GF}}(p^{m})$ holding an infinite family of 2-designs for any field ${\mathrm{ GF}}(q)$. In particular, some of these almost MDS codes are near MDS. Second, we present an infinite family of near MDS codes over ${\mathrm{ GF}}(2^{m})$ holding an infinite family of 3-designs by considering the number of roots of a special linearized polynomial. Compared to previous constructions of 3-designs or 2-designs from linear codes, the parameters of some of our designs are new and flexible. [ABSTRACT FROM AUTHOR]

Linear codes over $\mathbb {Z}_{p^{s}}$ of length $n$ are subgroups of $\mathbb {Z}_{p^{s}}^{n}$. These codes are also called $\mathbb {Z}_{p^{s}}$ -additive codes and can be seen as a generalization of linear codes over $\mathbb {Z}_{2}$ and $\mathbb {Z}_{4}$. A $\mathbb {Z}_{p^{s}}$ -linear code is a code over $\mathbb {Z}_{p}$ , not necessarily linear, which is the generalized Gray map image of a $\mathbb {Z}_{p^{s}}$ -additive code. In 2015, a systematic encoding was found for $\mathbb {Z}_{4}$ -linear codes. Moreover, an alternative permutation decoding method, which is suitable for any binary code (not necessarily linear) with a systematic encoding, was established. In this paper, we generalize these results by presenting a systematic encoding for any $\mathbb {Z}_{p^{s}}$ -linear code with $s\geq 2$ and $p$ prime. We also describe a permutation decoding method for any systematic code over $\mathbb {Z}_{p}$ , not necessarily linear, and show some examples of how to use this systematic encoding in this decoding method. [ABSTRACT FROM AUTHOR]

This paper presents the first systematic study of the zero-error capacity regions of noisy networks. First, we consider two simple such networks, each consisting of a stationary memoryless multiple access channel with two binary inputs and one discrete output. There are two users in each network. Each of the two users transmits a message through the network, and the sink(s) of the network can decode both messages with zero error. A graph is used to represent the distinguishability of the inputs of the channel, and a graph set is used to represent the distinguishability of the inputs of the network. We show that for two networks represented by the same graph set, their zero-error capacity regions are the same. We list all the possible graph sets for the two networks and determine the zero-error capacity regions for some of these graph sets. Based on this result, we explore a relation between graph theory and set theory, and then redefine the cancellative pair of families of subsets. We further extend the problem formulation to a general network called the parallel network, which may consist of more than one channel with multiple inputs and multiple outputs. [ABSTRACT FROM AUTHOR]

This paper investigates the problem of correcting multiple criss-cross insertions and deletions in arrays. More precisely, we study the unique recovery of $n \times n$ arrays affected by ${t}$ -criss-cross deletions defined as any combination of ${t_{\mathrm {r}}}$ row and ${t_{\mathrm {c}}}$ column deletions such that ${t_{\mathrm {r}}}+ {t_{\mathrm {c}}}= {t}$ for a given $t$. We show an equivalence between correcting ${t}$ -criss-cross deletions and ${t}$ -criss-cross insertions and show that a code correcting ${t}$ -criss-cross insertions/deletions has redundancy at least ${t} n + {t}\log n - \log ({t}!)$. Then, we present an existential construction of a ${t}$ -criss-cross insertion/deletion correcting code with redundancy bounded from above by ${t} n + \mathcal {O}({t}^{2} \log ^{2} n)$. The main ingredients of the presented code construction are systematic binary ${t}$ -deletion correcting codes and Gabidulin codes. The first ingredient helps locating the indices of the inserted/deleted rows and columns, thus transforming the insertion/deletion-correction problem into a row/column erasure-correction problem which is then solved using the second ingredient. [ABSTRACT FROM AUTHOR]

Sum-rank metric codes have recently attracted the attention of many researchers, due to their relevance in several applications. Mathematically, the sum-rank metric is a natural generalization of both the Hamming metric and the rank metric. In this paper, we provide an Anticode Bound for the sum-rank metric, which extends the corresponding Hamming and rank-metric Anticode bounds. We classify then optimal anticodes, i.e., codes attaining the sum-rank metric Anticode Bound. We use these optimal anticodes to define generalized sum-rank weights and we study their main properties. In particular, we prove that the generalized weights of an MSRD code are determined by its parameters. As an application, in the Appendix we explain how generalized weights measure information leakage in multishot network coding. [ABSTRACT FROM AUTHOR]

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

In order to accommodate the ever-growing data from various, possibly independent, sources and the dynamic nature of data usage rates in practical applications, modern cloud data storage systems are required to be scalable, flexible, and heterogeneous. The recent rise of the blockchain technology is also moving various information systems towards decentralization to achieve high privacy at low costs. While codes with hierarchical locality have been intensively studied in the context of centralized cloud storage due to their effectiveness in reducing the average reading time, those for decentralized storage networks (DSNs) have not yet been discussed. In this paper, we propose a joint coding scheme where each node receives extra protection through the cooperation with nodes in its neighborhood in a heterogeneous DSN with any given topology. This work extends and subsumes our prior work on coding for centralized cloud storage. In particular, our proposed construction not only preserves desirable properties such as scalability and flexibility, which are critical in dynamic networks, but also adapts to arbitrary topologies, a property that is essential in DSNs but has been overlooked in existing works. [ABSTRACT FROM AUTHOR]

In this article, we present a new construction of evaluation codes in the Hamming metric, which we call twisted Reed–Solomon codes. Whereas Reed–Solomon (RS) codes are MDS codes, this need not be the case for twisted RS codes. Nonetheless, we show that our construction yields several families of MDS codes. Further, for a large subclass of (MDS) twisted RS codes, we show that the new codes are not generalized RS codes. To achieve this, we use properties of Schur squares of codes as well as an explicit description of the dual of a large subclass of our codes. We conclude the paper with a description of a decoder, that performs very well in practice as shown by extensive simulation results. [ABSTRACT FROM AUTHOR]

In this paper, we mainly study quaternary linear codes and their binary subfield codes. First we obtain a general explicit relationship between quaternary linear codes and their binary subfield codes in terms of generator matrices and defining sets. Second, we construct quaternary linear codes via simplicial complexes and determine the weight distributions of these codes. Third, the weight distributions of the binary subfield codes of these quaternary codes are also computed by employing the general characterization. Furthermore, we present two infinite families of optimal linear codes with respect to the Griesmer Bound, and a class of binary almost optimal codes with respect to the Sphere Packing Bound. We also need to emphasize that we obtain at least 9 new quaternary linear codes. [ABSTRACT FROM AUTHOR]

Quantum LDPC codes are a promising direction for low overhead quantum computing. In this paper, we propose a generalization of the Union-Find decoder as a decoder for quantum LDPC codes. We prove that this decoder corrects all errors with weight up to $An^\alpha $ for some $A, \alpha > 0$ , where $n$ is the code length, for different classes of quantum LDPC codes such as toric codes and hyperbolic codes in any dimension $D \geq 3$ and quantum expander codes. To prove this result, we introduce a notion of covering radius which measures the spread of an error from its syndrome. We believe this notion could find application beyond the decoding problem. We also perform numerical simulations, which show that our Union-Find decoder outperforms the belief propagation decoder in the low error rate regime in the case of a quantum LDPC code with length 3600. [ABSTRACT FROM AUTHOR]

For the recently developed quasi-reversible BCH codes with long lengths and high error-correcting capability, this paper is aimed at proposing a new and faster decoding procedure. It consists of four steps: 1) compute the consecutive syndromes; 2) calculate the syndrome functions by the forward and backward recursions; 3) solve a linear subsystem together with one matrix multiplication in order to find an error-locator polynomial; 4) determine the errors from the obtained polynomial by using the root-finding algorithm. This procedure, especially in Steps 2 and 3, differs greatly from the conventional procedures, which determine an error-locator polynomial directly from solving a linear system with the aid of the consecutive syndromes. The key idea behind this decoding technique is that the computational complexity of such a small subsystem instead of an originally large linear system can be significantly reduced, although there are additional forward and backward syndrome calculations with low complexity increasing. Finally, the illustrative examples and numerical simulations can be helpful to demonstrate the accuracy and efficacy of the presented decoding technique at different error-correcting capabilities. [ABSTRACT FROM AUTHOR]

Let $ \mathbb K$ be an algebraically closed field. In this paper, we consider the class of smooth plane curves of degree $n+1>3$ over $ \mathbb K$ , containing three points, $P_{1},P_{2}$ , and $P_{3}$ , such that $nP_{1}+P_{2}$ , $nP_{2}+P_{3}$ , and $nP_{3}+P_{1}$ are divisors cut out by three distinct lines. For such curves, we determine the dimension of certain special divisors supported on $\{P_{1},P_{2},P_{3}\}$ , as well as an explicit description of all pure gaps at each nonempty subset of the distinguished points $P_{1},P_{2}$ , and $P_{3}$. When $ \mathbb K=\overline { \mathbb F}_{q}$ , this class of curves, which includes the Hermitian curve, is used to construct algebraic geometry codes having minimum distance better than the Goppa bound. [ABSTRACT FROM AUTHOR]

A $((k,n))$ quantum threshold secret sharing (QTS) scheme is a quantum cryptographic protocol for sharing a quantum secret among $n$ parties such that the secret can be recovered by any $k$ or more parties while $k-1$ or fewer parties have no information about the secret. Despite extensive research on these schemes, there has been very little study on optimizing the quantum communication cost during recovery. Recently, we initiated the study of communication efficient quantum threshold secret sharing (CE-QTS) schemes. These schemes reduce the communication complexity in QTS schemes by accessing $d>k$ parties for recovery; here $d$ is fixed ahead of encoding the secret. In contrast to the standard QTS schemes which require $k$ qudits for recovering each qudit in the secret, these schemes have a lower communication cost of $\frac {d}{d-k+1}$. In this paper, we further develop the theory of communication efficient quantum threshold schemes. Here, we propose universal CE-QTS schemes which reduce the communication cost for all $d>k$ simultaneously. We provide a framework based on ramp quantum secret sharing to construct CE-QTS and universal CE-QTS schemes. We give another construction for universal CE-QTS schemes based on Staircase codes. We derived a lower bound on communication complexity and show that our constructions are optimal. Finally, an information theoretic model is developed to analyse CE-QTS schemes and the lower bound on communication complexity is proved again using this model. [ABSTRACT FROM AUTHOR]

In this paper, we make a comprehensive study of two classes of Boolean functions whose interest originally comes from hybrid symmetric-FHE encryption (with stream ciphers like FiLIP), but which also present much interest for general stream ciphers. The functions in these two classes are cheap and easy to implement, and they allow the resistance to all classical attacks and to their guess and determine variants as well. We determine exactly all the main cryptographic parameters (algebraic degree, resiliency order, nonlinearity, algebraic immunity) for all functions in these two classes, and we give close bounds for the others (fast algebraic immunity, the dimension of the space of annihilators of minimal degree). This is the first time that this is done for all functions in large classes of cryptographic interest. [ABSTRACT FROM AUTHOR]

Minimal linear codes have significant applications in secret sharing schemes and secure two-party computation. When they are defined over finite fields, those codes have been intensively studied, especially in recent years, but they have been firstly partially characterized by Ashikhmin and Barg since 1998. Next, they were completely characterized in 2018 by Ding, Heng, and Zhou in terms of the minimum and maximum nonzero weights in the corresponding codes. Since then, many construction methods for minimal linear codes over finite fields throughout algebraic and geometric approaches have been proposed in the literature. In particular, the algebraic approach gives rise to minimal codes from (cryptographic) functions. Linear codes over finite fields have been expanded into the collection of acceptable alphabets for codes and study codes over finite commutative rings. A natural way to extend the known results available in the literature is to consider minimal linear codes over commutative rings with unity. In extending coding theory to codes over rings, several essential principles must be considered. Particularly extending the minimality property from finite fields to rings and creating such codes is not simple. Such an extension offers more flexibility in the construction of minimal codes. The present article investigates one-dimensional minimal linear codes over the rings $\mathbb {Z}_{p^{n}}$ (where $p$ is a prime) and $\mathbb {Z}_{p^{m}q^{n}}$ (where $p < q$ are distinct primes and $m\leq n$). Our ultimate objective is to characterize such codes’ minimality and design minimal linear codes over the considered rings. Given our objective, we first introduced the notion of minimal codes over (commutative) rings and succeeded in deriving simple characterization of one-dimensional minimal linear codes over the underlying rings mentioned above. Our new algebraic approach allows designing new minimal linear codes. Almost minimal codes over rings are also presented. To the best of our knowledge, the present paper offers a wide variety of minimal codes over (commutative) rings for the first time. Novel perspectives and developments in this direction are expected in the future. [ABSTRACT FROM AUTHOR]

This paper studies properties of binary runlength-limited sequences with additional constraints on their Hamming weight and/or their number of runs of identical symbols. An algebraic and a probabilistic (entropic) characterization of the exponential growth rate of the number of such sequences, i.e., their information capacity, are obtained by using the methods of multivariate analytic combinatorics, and properties of the capacity as a function of its parameters are stated. The second-order term in the asymptotic expansion of the rate of these sequences is also given, and the typical values of the relevant quantities are derived. Several applications of the results are illustrated, including bounds on codes for weight-preserving and run-preserving channels (e.g., the run-preserving insertion-deletion channel), a sphere-packing bound for channels with sparse error patterns, and the asymptotics of constant-weight sub-block constrained sequences. In addition, the asymptotics of a closely related notion— $q$ -ary sequences with fixed Manhattan weight—is briefly discussed, and an application in coding for molecular timing channels is illustrated. [ABSTRACT FROM AUTHOR]

Coded computation techniques provide robustness against straggling workers in distributed computing. However, most of the existing schemes require exact provisioning of the straggling behavior and ignore the computations carried out by straggling workers. Moreover, these schemes are typically designed to recover the desired computation results accurately, while in many machine learning and iterative optimization algorithms, faster approximate solutions are known to result in an improvement in the overall convergence time. In this paper, we first introduce a novel coded matrix-vector multiplication scheme, called coded computation with partial recovery (CCPR), which benefits from the advantages of both coded and uncoded computation schemes, and reduces both the computation time and the decoding complexity by allowing a trade-off between the accuracy and the speed of computation. We then extend this approach to distributed implementation of more general computation tasks by proposing a coded communication scheme with partial recovery, where the results of subtasks computed by the workers are coded before being communicated. Numerical simulations on a large linear regression task confirm the benefits of the proposed scheme in terms of the trade-off between the computation accuracy and latency. [ABSTRACT FROM AUTHOR]

We consider the $K$ -user Multiple Input Multiple Output (MIMO) Gaussian interference channel with $M$ antennas at each transmitter and $N$ antennas at each receiver. It is assumed that channel coefficients are constant real numbers and are available at all transmitters and at all receivers. The main objective of this paper is to characterize the number of Degrees of Freedom (DoF) of this channel. Using the real interference alignment technique introduced in Motahari et al., 2014, we show that $\frac {MN}{M+N} K$ degrees of freedom can be achieved for almost all channel realizations. Also, a new upper-bound on the DoF of this channel is provided. This upper-bound coincides with our achievable DoF for $K\geq K_{u} \triangleq \frac {M+N}{\gcd (M,N)}$ , where $\gcd (M,N)$ denotes the greatest common divisor of $M$ and $N$. This gives an exact characterization of DoF for $M\times N$ MIMO Gaussian interference channel in the case of $K\geq K_{u}$. Since there is no cooperation between transmit (or receive) antennas of each user in our transmission scheme, this result shows that the DoF benefit of joint processing in collocated antennas vanishes when the number of users is greater than a certain threshold. [ABSTRACT FROM AUTHOR]

Just as rank-metric or Gabidulin codes may be used to construct rate–diversity tradeoff optimal space–time codes, a recently introduced generalization for the sum-rank metric—linearized Reed–Solomon codes—accomplishes the same in the case of multiple fading blocks. In this paper, we provide the first explicit construction of minimal delay rate–diversity optimal multiblock space–time codes as an application of linearized Reed–Solomon codes. We also provide sequential decoders for these codes and, more generally, space–time codes constructed from finite field codes. Simulation results show that the proposed codes can outperform full diversity codes based on cyclic division algebras at low SNRs as well as utilize significantly smaller constellations. [ABSTRACT FROM AUTHOR]

A new family of codes, called clustering-correcting codes, is presented in this paper. This family of codes is motivated by the special structure of the data that is stored in DNA-based storage systems. The data stored in these systems has the form of unordered sequences, also called strands, and every strand is synthesized thousands to millions of times, where some of these copies are read back during sequencing. Due to the unordered structure of the strands, an important task in the decoding process is to place them in their correct order. This is usually accomplished by allocating part of the strand for an index. However, in the presence of errors in the index field, important information on the order of the strands may be lost. Clustering-correcting codes ensure that if the distance between the index fields of two strands is small, their data fields have large distance. It is shown how this property enables to place the strands together in their correct clusters even in the presence of errors. We present lower and upper bounds on the size of clustering-correcting codes and an explicit construction of these codes which uses only a single symbol of redundancy. The results are first presented for the Hamming metric and are then extended for the edit distance. [ABSTRACT FROM AUTHOR]

Reed-Muller codes are error-correcting codes used in many areas related to coding theory, such as electrical engineering and computer science. The binary $r^{th}$ -order Reed-Muller code $RM(r,n)$ can be viewed as the set of all $n$ -variable Boolean functions of algebraic degree at most $r$. Despite the intense work on these codes, many problems are known to be hard (notably, determining their covering radius) and remain open to this day. Fourteen years ago, Carlet and Mesnager improved in [IEEE Transactions on Information Theory, “Improving the Upper Bounds on the Covering Radii of Binary Reed-Muller Codes”, 53(1), 2007] the upper bound on the covering radius of the Reed-Muller code of order 2, and they deduced improved upper bounds on the covering radii of the Reed-Muller codes of higher orders. Until 2021, these upper bounds remained the best ones in the literature. The Reed-Muller code $RM(n-3,n)$ , which corresponds to the dual of the Reed-Muller code $RM(2,n)$ , has attracted much attention. One of the main reasons is that it is precisely the code that has been considered to get the upper bounds derived by Carlet and Mesnager. Those upper bounds have been obtained thanks to the characterization of the codewords of the Reed-Muller code, whose Hamming weights are strictly less than 2.5 times the minimum distance $2^{n-r}$ due to Kasami, Tokura, and Azumi. Despite their impressive work in the seventieth, a more refined study and profound description of those codewords of ${RM}({n-3},{n})$ whose Hamming weight equals 16, and especially 18, seem necessary, as it could help us significantly in improving the covering radius of Reed-Muller codes. In this paper, we push further the known results on the Reed-Muller codes by focusing on the Reed-Muller code ${RM}({n-3},{n})$. We provide a classification of the codewords of weights 16 and 18 of the Reed-Muller code $RM(n-3,n)$. Our algebraic descriptions allow us to count the number of such codewords and to enumerate all of them explicitly. [ABSTRACT FROM AUTHOR]

This paper employs auxiliary receivers as a mathematical tool to identify Gallager-type auxiliary random variables and write outer bounds for some basic multiuser settings. This approach is then applied to the relay, interference, and broadcast channel settings, yielding new outer bounds that improve on existing outer bounds and strictly outperform classical outer bounds. For instance, we strictly improve on: the cutset outer bound for the scalar Gaussian relay channel, the outer bounds for the Gaussian Z-interference channel, and the outer bounds for the two receiver broadcast channel. [ABSTRACT FROM AUTHOR]

As a special subclass of cyclic codes, BCH codes have wide applications in communication and storage systems. A BCH code of length $n$ over $\mathbb {F}_{q}$ is always relative to an $n$ -th primitive root of unity $\beta $ in an extension field of $\mathbb {F}_{q}$ , and is called a dually-BCH code if its dual is also a BCH code relative to the same $\beta $. The question as to whether a BCH code is a dually-BCH code is in general very hard to answer. In this paper, an answer to this question for primitive narrow-sense BCH codes and projective narrow-sense ternary BCH codes is given. Sufficient and necessary conditions in terms of the designed distances $\delta $ will be presented to ensure that these BCH codes are dually-BCH codes. In addition, the parameters of the primitive narrow-sense BCH codes and their dual codes are investigated. Some lower bounds on minimum distances of the dual codes of primitive and projective narrow-sense BCH codes are developed. Especially for binary primitive narrow-sense BCH codes, the new bounds on the minimum distances of the dual codes improve the classical Sidel’nikov bound, and are also better than the Carlitz and Uchiyama bound for large designed distances $\delta $. The question as to what subclasses of cyclic codes are BCH codes is also answered to some extent. As a byproduct, the parameters of some subclasses of cyclic codes are also investigated. [ABSTRACT FROM AUTHOR]

Diameter perfect codes form a natural generalization for perfect codes. They are based on the code-anticode bound which generalizes the sphere-packing bound. The code-anticode bound was proved by Delsarte for distance-regular graphs and it holds for some other metrics too. In this paper we prove the bound for non-binary constant-weight codes with the Hamming metric and characterize the diameter perfect codes and the maximum size anticodes for these codes. We distinguish between six families of non-binary diameter constant-weight codes and four families of maximum size non-binary constant-weight anticodes. Each one of these families of diameter perfect codes raises some different questions. We consider some of these questions and leave lot of ground for further research. Finally, as a consequence, some $t$ -intersecting families related to the well-known Erdös-Ko-Rado theorem, are constructed. [ABSTRACT FROM AUTHOR]

A functional PIR array code is a coding scheme which encodes some $s$ information bits into a $t\times m$ array such that every linear combination of the $s$ information bits has $k$ mutually disjoint recovering sets. Every recovering set consists of some of the array’s columns while it is allowed to read at most $\ell $ encoded bits from every column in order to receive the requested linear combination of the information bits. Functional batch array codes impose a stronger property where every multiset request of $k$ linear combinations has $k$ mutually disjoint recovering sets. Locality functional array codes demand that the size of every recovering set is restrained to be at most $r$. Given the values of $s, k, t, \ell,r$ , the goal of this paper is to study the optimal value of the number of columns $m$ such that these codes exist. Several lower bounds are presented as well as explicit constructions for several of these parameters. [ABSTRACT FROM AUTHOR]

Minimal linear codes are in one-to-one correspondence with special types of blocking sets of projective spaces over a finite field, which are called strong or cutting blocking sets. Minimal linear codes have been studied since decades but their tight connection with cutting blocking sets of finite projective spaces was unfolded only in the past few years, and it has not been fully exploited yet. In this paper we apply finite geometric and probabilistic arguments to contribute to the field of minimal codes. We prove an upper bound on the minimal length of minimal codes of dimension $k$ over the $q$ -element Galois field which is linear in both $q$ and $k$ , hence improve the previous superlinear bounds. This result determines the minimal length up to a small constant factor. We also improve the lower and upper bounds on the size of so called higgledy-piggledy line sets in projective spaces and apply these results to present improved bounds on the size of covering codes and saturating sets in projective spaces as well. [ABSTRACT FROM AUTHOR]

The deterministic identification (DI) capacity is developed in multiple settings of channels with power constraints. A full characterization is established for the DI capacity of the discrete memoryless channel (DMC) with and without input constraints. Originally, Ahlswede and Dueck established the identification capacity with local randomness at the encoder, resulting in a double exponential number of messages in the block length $n$. In the deterministic setup, the number of messages scales exponentially, as in Shannon’s transmission paradigm, but the achievable identification rates are higher. An explicit proof was not provided for the deterministic setting. In this paper, a detailed proof is presented for the DMC. Furthermore, Gaussian channels with fast and slow fading are considered, when channel side information is available at the decoder. A new phenomenon is observed as we establish that the number of messages scales as $2^{n\log (n)R}$ by deriving lower and upper bounds on the DI capacity on this scale. Consequently, the DI capacity of the Gaussian channel is infinite in the exponential scale and zero in the double exponential scale, regardless of the channel noise. [ABSTRACT FROM AUTHOR]

This paper considers the transmission of an infinite sequence of messages (a streaming source) over a packet erasure channel, where every source message must be recovered perfectly at the destination subject to a fixed decoding delay. While the capacity of a channel that introduces only bursts of erasures has been known for more than fifteen years, only recently, the capacity of a channel with either one burst of erasures or multiple arbitrary erasures in any fixed-sized sliding window has been established. However, explicit codes shown to achieve this capacity for any admissible set of parameters require a large field size that scales exponentially with the delay. Recently, it has been shown that there exist codes that achieve the capacity with field size which scales quadratically with the delay. However, explicit constructions were shown only to specific combinations of parameters. This work describes an explicit rate-optimal construction for all admissible channel and delay parameters over a field size that scales quadratically with the delay. [ABSTRACT FROM AUTHOR]

This paper proposes a new method of constructing compact fully-connected Quasi-Cyclic Low Density Parity Check (QC-LDPC) codes with girth $g$ = 8, 10, and 12. The originality of the proposed method is to impose constraints on the exponent matrix P to reduce the search space drastically. For a targeted lifting degree of $N$ , the first step of the method is to sieve the integer ring $\mathbb {Z}_{N}$ to make a particular sub-group with specific properties to construct the second column of P (the first column being filled with zeros). The remaining columns of P are determined recursively as multiples of the second column by adapting the sequentially multiplied column (SMC) method whereby a controlled greedy search is applied at each step. The codes constructed with the proposed semi-algebraic method show lengths that can be significantly shorter than their best counterparts in the literature. [ABSTRACT FROM AUTHOR]

Let $n (>3)$ be a prime number and ${\mathbb {F}}_{2^{n}}$ a finite field of $2^{n}$ elements. Let $L ={\mathbb {F}}_{2^{n}}\cup \{\infty \}$ be the support set and $g(x)$ an irreducible polynomial of degree 6 over ${\mathbb {F}}_{2^{n}}$. In this paper, we obtain an upper bound on the number of extended irreducible binary Goppa codes $\Gamma (L, g)$ of degree 6 and length $2^{n}+1$. [ABSTRACT FROM AUTHOR]

Let $q=p^{e}$ be a prime power and $\ell $ be an integer with $0\leq \ell \leq e-1$. The $\ell $ -Galois hull of classical linear codes is a generalization of the Euclidean hull and Hermitian hull. We provide a necessary and sufficient condition under which a codeword of a GRS code or an extended GRS code belongs to its $\ell $ -Galois dual code, generalizing both the Euclidean case and Hermitian case in the literature. By using four different tools: 1) the norm mapping from $\mathbb {F}_{q}^{\ast }$ to $\mathbb {F}_{p^{\ell }}^{\ast }$ ; 2) the direct product of two cyclic subgroups; 3) the coset decomposition of a cyclic group; 4) an additive subgroup of $\mathbb {F}_{q}$ and its cosets, we construct eleven families of $q$ -ary MDS codes with $\ell $ -Galois hulls of arbitrary dimensions, and give the related eleven families of $[[n,k,d;c]]_{q}$ entanglement-assisted quantum error-correcting codes (EAQECCs) with relatively large minimum distance in the sense that $2d=n-k+2+c$. We show that developing the theory on $\ell $ -Galois hulls of $q$ -ary MDS codes in this paper enables us to obtain new $q$ -ary EAQECCs with different kinds of length sets via different $\ell $ , where $2\ell \mid e$. [ABSTRACT FROM AUTHOR]

This paper considers “ $\delta $ -almost Reed–Muller codes”, i.e., linear codes spanned by evaluations of all but a $\delta $ fraction of monomials of degree at most $d$. It is shown that for any $\delta > 0$ and any $\varepsilon >0$ , there exists a family of $\delta $ -almost Reed–Muller codes of constant rate that correct $1/2- \varepsilon $ fraction of random errors with high probability. For exact Reed–Muller codes, the analogous result is not known and represents a weaker version of the longstanding conjecture that Reed–Muller codes achieve capacity for random errors (Abbe-Shpilka-Wigderson STOC ’15). Our proof is based on the recent polarization result for Reed–Muller codes, combined with a combinatorial approach to establishing inequalities between the Reed–Muller code entropies. [ABSTRACT FROM AUTHOR]

A large class of MDS linear codes is constructed. These codes are endowed with an efficient decoding algorithm. Both the definition of the codes and the design of their decoding algorithm only require from Linear Algebra methods, making them fully accessible for everyone. Thus, the first part of the paper develops a direct presentation of the codes by means of parity-check matrices, and the decoding algorithm rests upon matrix and linear maps manipulations. The somewhat more sophisticated mathematical context (non-commutative rings) needed for the proof of the correctness of the decoding algorithm is postponed to the second part. A final section locates the Reed-Solomon skew-differential codes introduced here within the general context of codes defined by means of skew polynomial rings. [ABSTRACT FROM AUTHOR]

Interleaved Reed–Solomon codes admit efficient decoding algorithms which correct burst errors far beyond half the minimum distance in the random errors regime, e.g., by computing a common solution to the Key Equation for each Reed–Solomon code, as described by Schmidt et al. If this decoder does not succeed, it may either fail to return a codeword or miscorrect to an incorrect codeword, and good upper bounds on the fraction of error matrices for which these events occur are known. The decoding algorithm immediately applies to interleaved alternant codes as well, i.e., the subfield subcodes of interleaved Reed–Solomon codes, but the fraction of decodable error matrices differs, since the error is now restricted to a subfield. In this paper, we present new general lower and upper bounds on the fraction of error matrices decodable by Schmidt et al.’s decoding algorithm, thereby making it the only decoding algorithm for interleaved alternant codes for which such bounds are known. [ABSTRACT FROM AUTHOR]

The de Bruijn graph, its sequences, and their various generalizations, have found many applications in information theory, including many new ones in the last decade. In this paper, motivated by a coding problem for emerging memory technologies, a set of sequences which generalize the window property of de Bruijn sequences, on its shorter subsequences, are defined. These sequences can be also defined and viewed as constrained sequences. Hence, they will be called locally-constrained de Bruijn sequences and a set of such sequences will be called a locally-constrained de Bruijn code. Several properties and alternative definitions for such codes are examined and they are analyzed as generalized sequences in the de Bruijn graph (and its generalization) and as constrained sequences. Various enumeration techniques are used to compute the total number of sequences for any given set of parameters. A construction method of such codes from the theory of shift-register sequences is proposed. Finally, we show how these locally-constrained de Bruijn sequences and codes can be applied in constructions of codes for correcting synchronization errors in the $\ell $ -symbol read channel and in the racetrack memory channel. For this purpose, these codes are superior in their size to previously known codes. [ABSTRACT FROM AUTHOR]

This paper presents finite-blocklength achievability bounds for the Gaussian multiple access channel (MAC) and random access channel (RAC) under average-error and maximal-power constraints. Using random codewords uniformly distributed on a sphere and a maximum likelihood decoder, the derived MAC bound on each transmitter’s rate matches the MolavianJazi-Laneman bound (2015) in its first- and second-order terms, improving the remaining terms to $\frac {1}2\frac {\log {n}}{n}+{O} \left ({\frac {1}{n}}\right)$ bits per channel use. The result $\vphantom {\sum ^{R}}$ then extends to a RAC model in which neither the encoders nor the decoder knows which of ${K}$ possible transmitters are active. In the proposed rateless coding strategy, decoding occurs at a time ${n}_{t}$ that depends on the decoder’s estimate ${t}$ of the number of active transmitters ${k}$. Single-bit feedback from the decoder to all encoders at each potential decoding time ${n}_{i}$ , ${i} \leq {t}$ , informs the encoders when to stop transmitting. For this RAC model, the proposed code achieves the same first-, second-, and third-order performance as the best known result for the Gaussian MAC in operation. [ABSTRACT FROM AUTHOR]

In this paper, we consider the list decoding property of codes under insertion and deletion errors (insdel for short). Firstly, we analyse the list decodability of random insdel codes. Our result provides a more complete picture on the list decodability of insdel codes when both insertion and deletion errors happen. Secondly, we construct a family of insdel codes along with their efficient encoding and decoding algorithms through concatenation method which provides a Zyablov-type bound for insdel metric codes. [ABSTRACT FROM AUTHOR]

The distributed matrix multiplication problem with an unknown number of stragglers is considered, where the goal is to efficiently and flexibly obtain the product of two massive matrices by distributing the computation across $N$ servers. There are up to $N - R$ stragglers but the exact number is not known a priori. Motivated by reducing the computation load of each server, a flexible solution is proposed to fully utilize the computation capability of available servers. The computing task for each server is separated into several subtasks, constructed based on Entangled Polynomial codes by Yu et al. The final results can be obtained from either a larger number of servers with a smaller amount of computation completed per server or a smaller number of servers with a larger amount of computation completed per server. The required finite field size of the proposed solution is less than $2N$. Moreover, the optimal design parameters such as the partitioning of the input matrices are discussed. Our constructions can also be generalized to other settings such as batch distributed matrix multiplication and secure distributed matrix multiplication. [ABSTRACT FROM AUTHOR]

We consider the repair problem for Reed–Solomon (RS) codes, evaluated on an $\mathbb {F}_{q}$ -linear subspace $U\subseteq \mathbb {F}_{q^{m}} $ of dimension $d$ , where $q$ is a prime power, $m$ is a positive integer, and $\mathbb {F}_{q}$ is the Galois field of size $q$. For $q>2$ , we show the existence of a linear repair scheme for the RS code of length $n=q^{d}$ and codimension $q^{s}$ , $s < d$ , evaluated on $U$ , in which each of the $n-1$ surviving nodes transmits only $r$ symbols of $\mathbb {F}_{q}$ , provided that $ms\geq d(m-r)$. For the case $q=2$ , we prove a similar result, with some restrictions on the evaluation linear subspace $U$. Our proof is based on a probabilistic argument, however the result is not merely an existence result; the success probability is fairly large (at least $1/3$) and there is a simple criterion for checking the validity of the randomly chosen linear repair scheme. Our result extend the construction of Dau–Milenkovic to the range $r < m-s$ , for a wide range of parameters. [ABSTRACT FROM AUTHOR]

We introduce two notions of discrepancy between binary vectors, which are not metric functions in general but nonetheless capture the mathematical structure of the binary asymmetric channel. These lead to two new fundamental parameters of binary error-correcting codes, both of which measure the probability that the maximum likelihood decoder fails. We then derive various bounds for the cardinality and weight distribution of a binary code in terms of these new parameters, giving examples of codes meeting the bounds with equality. [ABSTRACT FROM AUTHOR]