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In this paper, we provide two complementary results for cyclic codes of composite length. First, we give a general construction of cyclic codes of length nr from cyclic codes of length n and a lower bound on the minimum distance. Numerical data show that many cyclic codes of composite length with the best parameters can be obtained in this way. Second, in the other direction, we show that for cyclic codes of length n with a primitive n-th root of unity as a non-zero, the minimum distance is roughly bounded by the square-free part of n. This means that we shall not expect good cyclic codes of length n in general if, for example, the length n is divisible by a high power of a prime. [ABSTRACT FROM AUTHOR]

In this paper, we extend two earlier works further in two directions and compute the weight distribution of these cyclic codes under more relaxed conditions. It is interesting to note that many cyclic codes in the family are optimal and have only a few non-zero weights. Besides using similar ideas, we carry out some subtle manipulation of certain exponential sums. [ABSTRACT FROM AUTHOR]

In this paper, we address the computational hardness of optimally decoding a quantum stabilizer code. Much like classical linear codes, errors are detected by measuring certain check operators which yield an error syndrome, and the decoding problem consists of determining the most likely recovery given the syndrome. The corresponding classical problem is known to be NP-complete, and are appropriate a similar decoding problem for quantum codes is also known to be NP-complete. However, this decoding strategy is not optimal in the quantum setting as it does not consider error degeneracy, which causes distinct errors to have the same effect on the code. Here, we show that optimal decoding of stabilizer codes (previously known to be NP-hard) is in fact computationally much harder than optimal decoding of classical linear codes, it is #P-complete. [ABSTRACT FROM AUTHOR]

Cyclic codes are an interesting class of linear codes due to their efficient encoding and decoding algorithms. Bose-Ray-Chaudhuri-Hocquenghem (BCH) codes form a subclass of cyclic codes and are very important in both theory and practice as they have good error-correcting capability and are widely used in communication systems, storage devices, and consumer electronics. However, the dimension and minimum distance of BCH codes are not known in general. The objective of this paper is to determine the Bose and minimum distances of a class of narrow-sense primitive BCH codes. [ABSTRACT FROM AUTHOR]

It was shown by Massey that linear complementary dual (LCD) codes are asymptotically good. In 2004, Sendrier proved that LCD codes meet the asymptotic Gilbert–Varshamov (GV) bound. Until now, the GV bound still remains to be the best asymptotical lower bound for LCD codes. In this paper, we show that an algebraic geometry code over a finite field of even characteristic is equivalent to an LCD code and consequently there exists a family of LCD codes that are equivalent to algebraic geometry codes and exceed the asymptotical GV bound. [ABSTRACT FROM AUTHOR]

Linear complementary-dual (LCD for short) codes are linear codes that intersect with their duals trivially. LCD codes have been used in certain communication systems. It is recently found that LCD codes can be applied in cryptography. This application of LCD codes renewed the interest in the construction of LCD codes having a large minimum distance. Maximum distance separable (MDS) codes are optimal in the sense that the minimum distance cannot be improved for given length and code size. Constructing LCD MDS codes is thus of significance in theory and practice. Recently, Jin constructed several classes of LCD MDS codes through generalized Reed-Solomon codes. In this paper, a different approach is proposed to obtain new LCD MDS codes from generalized Reed-Solomon codes. Consequently, new code constructions are provided and certain previously known results by Jin are extended. [ABSTRACT FROM AUTHOR]

Motivated by applications in DNA-based storage, we introduce the new problem of code design in the Damerau metric. The Damerau metric is a generalization of the Levenshtein distance which, in addition to deletions, insertions, and substitution errors also accounts for adjacent transposition edits. We first provide constructions for codes that may correct either a single deletion or a single adjacent transposition and then proceed to extend these results to codes that can simultaneously correct a single deletion and multiple adjacent transpositions. We conclude with constructions for joint block deletion and adjacent block transposition error-correcting codes.11

Parts of the results were presented at the International Symposium on Information Theory in Barcelona, 2016.

Expander codes are Tanner codes defined on sparse graphs that have good expansion properties. Sipser and Spielman (1996) showed that there is a linear-time decoding algorithm for expander codes when the vertex expansion is at least 3/4 and the number of errors corrected is a constant fraction of the code length. Later, Feldman et al. (2007) gave a decoding algorithm that allows the expansion to be 2/3 + 1/(3c), where $c$ is the left degree of the underlying bipartite graph, at the expense of polynomial-time decoding complexity. Recently, Viderman (2013) further improved the expansion parameter to $2/3 - 1/(6c)$ , and the decoding algorithm runs in linear time. These results are for expander codes whose inner codes are parity-check codes. By using stronger inner codes, Chilappagari et al. (2010) showed that there is a linear-time decoding algorithm for every vertex expansion greater than 1/2. In this paper, it is shown that for every vertex expansion, there is a linear-time decoding algorithm for expander codes (using inner codes with minimum distance depending on the vertex expansion), and that the number of errors corrected is a constant fraction of the code length. [ABSTRACT FROM AUTHOR]

Reducible codes for the rank metric were introduced for cryptographic purposes. They have fast encoding and decoding algorithms, include maximum rank distance (MRD) codes, and can correct many rank errors beyond half of their minimum rank distance, which makes them suitable for error correction in network coding. In this paper, we study their security behavior against information leakage on networks when applied as coset coding schemes, giving the following main results: 1) we give lower and upper bounds on their generalized rank weights (GRWs), which measure worst case information leakage to the wire tapper; 2) we find new parameters for which these codes are MRD (meaning that their first GRW is optimal) and use the previous bounds to estimate their higher GRWs; 3) we show that all linear (over the extension field) codes, whose GRWs are all optimal for fixed packet and code sizes but varying length are reducible codes up to rank equivalence; and 4) we show that the information leaked to a wire tapper when using reducible codes is often much less than the worst case given by their (optimal in some cases) GRWs. We conclude with some secondary related properties: conditions to be rank equivalent to Cartesian products of linear codes and conditions to be rank degenerate, duality properties, and MRD ranks. [ABSTRACT FROM PUBLISHER]

In this paper we study stopping sets of AG codes over a family of curves, denoted by Xƒ,g, that includes several important curves with applications in coding theory. We present results concerning stopping sets of one-point and m-point codes over Xƒ,g , generalizing some results for Hermitian codes presented by Anderson and Matthews. [ABSTRACT FROM AUTHOR]

Generalized low-density parity-check (GLDPC) codes are a class of LDPC codes in which the standard single parity check (SPC) constraints are replaced by constraints defined by a linear block code. These stronger constraints typically result in improved error floor performance, due to better minimum distance and trapping set properties, at a cost of some increased decoding complexity. In this paper, we study spatially coupled generalized low-density parity-check (SC-GLDPC) codes and present a comprehensive analysis of these codes, including: (1) an iterative decoding threshold analysis of SC-GLDPC code ensembles demonstrating capacity approaching thresholds via the threshold saturation effect; (2) an asymptotic analysis of the minimum distance and free distance properties of SC-GLDPC code ensembles, demonstrating that the ensembles are asymptotically good; and (3) an analysis of the finite-length scaling behavior of both GLDPC block codes and SC-GLDPC codes based on a peeling decoder (PD) operating on a binary erasure channel (BEC). Results are compared to GLDPC block codes, and the advantages and disadvantages of SC-GLDPC codes are discussed. [ABSTRACT FROM AUTHOR]

In this paper, we first study in detail the relationship between minimal linear codes and cutting blocking sets, recently introduced by Bonini and Borello, and then completely characterize minimal linear codes as cutting blocking sets. As a direct result, minimal projective codes of dimension 3 and t-fold blocking sets with t ≥ 2 in projective planes are identical objects. Some bounds on the parameters of minimal codes are derived from this characterization. Using this new link between minimal codes and blocking sets, we also present new general primary and secondary constructions of minimal linear codes. As a result, infinite families of minimal linear codes not satisfying the Aschikhmin-Barg’s condition are obtained. In addition to this, open problems on the parameters and the weight distributions of some generated linear codes are presented. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of linear codes defined by characteristic functions of certain subsets of a finite field. We derive a sufficient and necessary condition for such a code to be a minimal linear code by a character-theoretical approach. We obtain new three-weight or four-weight minimal linear codes that do not satisfy the Ashikhmin-Barg condition by using partial difference sets. We show that our construction yields minimal linear codes that do not arise from cutting vectorial blocking sets, and also discuss their applications in secret sharing schemes. [ABSTRACT FROM AUTHOR]

Consider two remote nodes (encoder and decoder), each with a binary sequence. The encoder’s sequence X differs from the decoder’s sequence Y by a small number of edits (deletions and insertions). The goal is to construct a message M, to be sent via a one-way error free link, such that the decoder can reconstruct X using M and Y. In this paper, we devise a coding scheme for this one-way synchronization model. The scheme is based on multiple layers of Varshamov-Tenengolts (VT) codes combined with off-the-shelf linear error-correcting codes, and uses a list decoder. We bound the expected list size of the decoder under certain assumptions, and validate its performance via numerical simulations. We also consider an alternative decoder that uses only the constraints from the VT codes (i.e., does not require a linear code), and has a smaller redundancy at the expense of a slightly larger average list size. [ABSTRACT FROM AUTHOR]

This paper develops a fundamental theory of realizations of linear and group codes on general graphs using elementary group theory, including basic group duality theory. Principal new and extended results include: normal realization duality; analysis of systems-theoretic properties of fragments of realizations and their connections; minimal \(\Leftrightarrow \) trim and proper theorem for cycle-free codes; results showing that all constraint codes except interface nodes may be assumed to be trim and proper, and that the interesting part of a cyclic realization is its 2-core; notions of observability and controllability for fragments, and related tests; and relations between state-trimness and controllability, and dual state-trimness and observability. [ABSTRACT FROM AUTHOR]

The class of \(q\) -ary list permutation invariant linear codes is introduced in this paper along with probabilistic arguments that validate their existence when certain conditions are met. The specific class of codes is characterized by an upper bound that is tighter than the generalized Shulman–Feder bound and relies on the distance of the codes’ weight distribution to the binomial (multinomial, respectively) one. The bound applies to cases where a code from the proposed class is transmitted over a \(q\) -ary output symmetric discrete memoryless channel and list decoding with fixed list size is performed at the output. In the binary case, the new upper bounding technique allows the discovery of list permutation invariant codes whose upper bound coincides with sphere-packing exponent. Furthermore, the proposed technique motivates the introduction of a new class of upper bounds for general \(q\) -ary linear codes whose members are at least as tight as the DS2 bound as well as all its variations for the discrete channels treated in this paper. [ABSTRACT FROM AUTHOR]

Erasure list decoding was introduced to correct a larger number of erasures by outputting a list of possible candidates. In this paper, we consider both random linear codes and algebraic geometry codes for list decoding from erasures. The contributions of this paper are twofold. First, for arbitrary \(00\) ( \(R\) and \(\epsilon \) are independent), we show that with high probability a \(q\) -ary random linear code of rate \(R\) is an erasure list-decodable code with constant list size \(q^{O(1/\epsilon )}\) that can correct a fraction \(1-R-\epsilon \) of erasures, i.e., a random linear code achieves the information-theoretic optimal tradeoff between information rate and fraction of erasures. Second, we show that algebraic geometry codes are good erasure list-decodable codes. Precisely speaking, a \(q\) -ary algebraic geometry code of rate \(R\) from the Garcia-Stichtenoth tower can correct \(1-R-({1}/{\sqrt {q}-1})+({1}/{q})-\epsilon \) fraction of erasures with list size \(O(1/\epsilon )\) . This improves the Johnson bound for erasures applied to algebraic geometry codes. Furthermore, list decoding of these algebraic geometry codes can be implemented in polynomial time. Note that the code alphabet size \(q\) in this paper is constant and independent of \(\epsilon \) . [ABSTRACT FROM PUBLISHER]

Binary codes over 2D arrays are very useful in data storage, where each array column represents a storage device or unit that may suffer failure. In this paper, we propose a new framework for probabilistic construction of codes on 2D arrays. Instead of a pure combinatorial erasure model used in traditional array codes, we propose a mixed combinatorial-probabilistic model of limiting the number of column failures, and assuming a binary erasure channel in each failing column. For this model, we give code constructions and detailed analysis that allow sustaining a large number of column failures with graceful degradation in the fraction of erasures correctable in failing columns. Another advantage of the new framework is that it uses low-complexity iterative decoding. The key component in the analysis of the new codes is to analyze the decoding graphs induced by the failed columns, and infer the decoding performance as a function of the code design parameters, as well as the array size and failure parameters. A particularly interesting class of codes, called probabilistically maximum distance separable (MDS) array codes, gives fault-tolerance that is equivalent to traditional MDS array codes. The results also include a proof that the 2D codes outperform standard 1D low-density parity-check codes. [ABSTRACT FROM PUBLISHER]

In this paper, we estimate the fidelity of stabilizer and CSS codes. First, we derive a lower bound on the fidelity of a stabilizer code via its quantum enumerator. Next, we find the average quantum enumerators of the ensembles of finite length stabilizer and CSS codes. We use the average quantum enumerators for obtaining lower bounds on the average fidelity of these ensembles. We further improve the fidelity bounds by estimating the quantum enumerators of expurgated ensembles of stabilizer and CSS codes. Finally, we derive fidelity bounds in the asymptotic regime when the code length tends to infinity. These results tell us which code rate we can afford for achieving a target fidelity with codes of a given length. The results also show that in symmetric depolarizing channel a typical stabilizer code has better performance, in terms of fidelity and code rate, compared with a typical CSS codes, and that balanced CSS codes significantly outperform other CSS codes. Asymptotic results demonstrate that CSS codes have a fundamental performance loss compared with stabilizer codes. [ABSTRACT FROM PUBLISHER]

It has been a great challenge to construct new quantum maximum-distance-separable (MDS) codes. In particular, it is very hard to construct the quantum MDS codes with relatively large minimum distance. So far, except for some sparse lengths, all known q -ary quantum MDS codes have minimum distance {\leq}q/2+1 . In this paper, we provide a construction of the quantum MDS codes with minimum distance {>}q/2+1 . In particular, we show the existence of the q -ary quantum MDS codes with length n=q^{2}+1 and minimum distance d for any d\leq q+1 (this result extends those given in [10], [12], and [13]); and with length (q^{2}+2)/3 and minimum distance d$ for any d\leq (2q+2)/3$ if 3\vert (q+1)$ . Our method is through Hermitian self-orthogonal codes. The main idea of constructing the Hermitian self-orthogonal codes is based on the solvability in of a system of homogenous equations over {\BBF_{q^{2}}} . [ABSTRACT FROM PUBLISHER]

It is known that the empirical spectral distribution of random matrices obtained from linear codes of increasing length converges to the well-known Marchenko-Pastur law, if the Hamming distance of the dual codes is at least 5. In this paper, we prove that the convergence rate in probability is at least of the order $n^{-1/4}$ where $n$ is the length of the code. [ABSTRACT FROM AUTHOR]

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

A locally repairable code (LRC) is a linear code such that every code symbol can be recovered by accessing a small number of other code symbols. In this paper, we study bounds and constructions of LRCs from the viewpoint of parity-check matrices. Firstly, a simple and unified framework based on parity-check matrix to analyze the bounds of LRCs is proposed, and several new explicit bounds on the minimum distance of LRCs in terms of the field size are presented. In particular, we give an alternate proof of the Singleton-like bound for LRCs first proved by Gopalan et al. Some structural properties on optimal LRCs that achieve the Singleton-like bound are given. Then, we focus on constructions of optimal LRCs over the binary field. It is proved that there are only five classes of possible parameters with which optimal binary LRCs exist. Moreover, by employing the proposed parity-check matrix approach, we completely enumerate all these five classes of optimal binary LRCs attaining the Singleton-like bound in the sense of equivalence of linear codes. [ABSTRACT FROM AUTHOR]

An index code for broadcast channel with receiver side information is locally decodable if each receiver can decode its demand by observing only a subset of the transmitted codeword symbols instead of the entire codeword. Local decodability in index coding is known to reduce receiver complexity, improve user privacy and decrease decoding error probability in wireless fading channels. Conventional index coding solutions assume that the receivers observe the entire codeword, and as a result, for these codes the number of codeword symbols queried by a user per decoded message symbol, which we refer to as locality, could be large. In this paper, we pose the index coding problem as that of minimizing the broadcast rate for a given value of locality (or vice versa) and designing codes that achieve the optimal trade-off between locality and rate. We identify the optimal broadcast rate corresponding to the minimum possible value of locality for all single unicast problems. We present new structural properties of index codes which allow us to characterize the optimal trade-off achieved by: vector linear codes when the side information graph is a directed cycle; and scalar linear codes when the minrank of the side information graph is one less than the order of the problem. We also identify the optimal trade-off among all codes, including non-linear codes, when the side information graph is a directed 3-cycle. Finally, we present techniques to design locally decodable index codes for arbitrary single unicast problems and arbitrary values of locality. [ABSTRACT FROM AUTHOR]

Locally repairable codes (LRCs) and codes with hierarchical locality (H-LRCs) have a great significance due to their applications in distributed storage systems. Constructing LRCs or H-LRCs achieving the Singleton-type bound is a challenging task and has received much attention. In this paper, we make use of optimal LRCs to construct a family of optimal H-LRCs. For this purpose we first construct a new class of optimal LRCs via generalized Reed-Solomon codes (GRS codes). Next, based on these optimal LRCs we present some new optimal H-LRCs, of which the parameters are more flexible than those given in and. [ABSTRACT FROM AUTHOR]

$\epsilon $ -Minimum Storage Regenerating ($\epsilon $ -MSR) codes form a special class of Maximum Distance Separable (MDS) codes, providing mechanisms for exact regeneration of a single code block in their codewords by downloading slightly sub-optimal amount of information from the remaining code blocks. The key advantage of these codes is a significantly lower sub-packetization that grows only logarithmically with the code length, while providing optimality in storage and error-correcting capacity. However, existing constructions of $\epsilon $ -MSR codes require each remaining code block to be available for the repair of any failed code block. In this paper, we construct $\epsilon $ -MSR codes that can repair any failed code block by contacting fewer number of available code blocks. When a code block fails, our repair procedure needs to contact a few compulsory code blocks and is free to choose any subset of available code blocks for the remaining choices. Our construction requires a field size linear in code length and ensures load balancing (in terms of information downloaded) among the contacted code blocks for repairing a failed code block. [ABSTRACT FROM AUTHOR]

The notion of Generalized Hamming weights of block codes has been investigated since the nineties due to its significant role in coding theory and cryptography. In this paper we extend this concept to the context of convolutional codes. In particular, we focus on column distances and introduce the novel notion of generalized column distances (GCD). We first show that the hierarchy of GCD is strictly increasing. We then provide characterizations of such distances in terms of the truncated parity-check matrix of the code, that will allow us to determine their values. Finally, the case in which the parity-check matrix is in systematic form is treated. [ABSTRACT FROM AUTHOR]

A Locally Recoverable Code is a code such that the value of any single coordinate of a codeword can be recovered from the values of a small subset of other coordinates. When we have $\delta $ non-overlapping subsets of cardinality $r_{i}$ that can be used to recover the missing coordinate we say that a linear code $\mathcal {C}$ with length $n$ , dimension $k$ , minimum distance $d$ has $(r_{1},\ldots, r_\delta)$ -locality and denote by $[n, k, d; r_{1}, r_{2}, {\dots }, r_\delta]$. In this paper we provide a new upper bound for the minimum distance of these codes. Working with a finite number of subgroups of cardinality $r_{i}+1$ of the automorphism group of a function field $\mathcal {F}| \mathbb {F}_{q}$ of genus $g \geq 1$ we propose a construction of $[n, k, d; r_{1}, r_{2}, {\dots }, r_\delta]$ -codes and apply the results to some well known families of function fields. [ABSTRACT FROM AUTHOR]

Self-orthogonal codes are an interesting type of linear codes due to their wide applications in communication and cryptography. It is known that self-orthogonal codes are often used to construct quantum error-correcting codes, which can protect quantum information in quantum computations and quantum communications. Let $\mathcal C$ be an $[n, k]$ cyclic code over $\mathbb {F}_{q}$ , where $\mathbb {F}_{q}$ is the finite field of order $q$. The hull of $\mathcal C$ is defined to be the intersection of the code and its dual. In this paper, we will employ the defining sets of cyclic codes to present two general characterizations of the hulls that have dimension $k-1$ or $k^\perp -1$ , where $k^\perp $ is the dimension of the dual code $\mathcal C^\perp $. Several sufficient and necessary conditions for primitive and projective BCH codes to have $(k-1)$ -dimensional (or $(k^\perp -1)$ -dimensional) hulls are also developed by presenting lower and upper bounds on their designed distances. Furthermore, several classes of self-orthogonal codes are proposed via the hulls of BCH codes and their parameters are also investigated. The dimensions and minimum distances of some self-orthogonal codes are determined explicitly. In addition, several optimal codes are obtained. [ABSTRACT FROM AUTHOR]

Codes over Galois rings have been studied extensively during the last three decades. Negacyclic codes over \mathop \mathrm GR\nolimits (2^a,m) of length 2^s have been characterized: the ring \mathcal R2(a,m,-1)= {\mathrm{ GR}}(2^{a},m)[x]/\langle x^{2^{s}}+1\rangle is a chain ring. Furthermore, these results have been generalized to \lambda -constacyclic codes for any unit \lambda of the form 4z-1 , . In this paper, we study more general cases and investigate all cases, where {\mathcal{ R}}_{p}(a,m,\gamma) = {\mathrm{ GR}}(p^{a},m)[x]/\langle x^{p^{s}}-\gamma \rangle is a chain ring. In particular, the necessary and sufficient conditions for the ring {\mathcal{ R}}_{p}(a,m,\gamma) to be a chain ring are obtained. In addition, by using this structure we investigate all \gamma -constacyclic codes over \mathop {\mathrm {GR}}\nolimits (p^{a},m) when {\mathcal{ R}}_{p}(a,m,\gamma)$ is a chain ring. The necessary and sufficient conditions for the existence of self-orthogonal and self-dual \gamma$ -constacyclic codes are also provided. Among others, for any prime p$ , the structure of is used to establish the Hamming and homogeneous distances of $\gamma$ -constacyclic codes. [ABSTRACT FROM PUBLISHER]

Historically, LCD cyclic codes were referred to as reversible cyclic codes, which had applications in data storage. Due to a newly discovered application in cryptography, there has been renewed interest in LCD codes. In this paper, we explore two special families of LCD cyclic codes, which are both BCH codes. The dimensions and the minimum distances of these LCD BCH codes are investigated. [ABSTRACT FROM AUTHOR]

A long-standing conjecture of Golomb and Welch, raised in 1970, states that there is no perfect r error correcting Lee code of length n for n\geq 3 and r>1 under the l_{p} metric, where 1\leq p<\infty . We show some nonexistence results of linear perfect lp codes for p=1 and 2\leq p<\infty , r=2^{1/p},3^{1/p} . We also give an algebraic construction of quasi-perfect l_{p}$ codes for p=1, r=2$ , and 2\leq p<\infty , r=2^{1/p} . [ABSTRACT FROM PUBLISHER]

In this paper, a streaming transmission setup is considered, where an encoder observes a new message in the beginning of each block and a decoder sequentially decodes each message after a delay of $T$ blocks. In this streaming setup, the fundamental interplay between the coding rate, the error probability, and the blocklength in the moderate deviations regime is studied. For output symmetric channels, the moderate deviations constant is shown to improve over the block coding or non-streaming setup by exactly a factor of $T$ for a certain range of moderate deviations scalings. For the converse proof, a more powerful decoder, to which some extra information is fedforward is assumed. The error probability is bounded first for an auxiliary channel and this result is translated back to the original channel by using a newly developed change-of-measure lemma, where the speed of decay of the remainder term in the exponent is carefully characterized. For the achievability proof, a known coding technique that involves a joint encoding and decoding of fresh and past messages is applied with some manipulations in the error analysis. [ABSTRACT FROM AUTHOR]

In this paper, we solve the open problems raised in [8] and present some examples to illustrate the obtained results. Moreover, we work out the diophantine problem by Shi and Wang and then give the sufficient conditions for the nonexistence of two-Lee weight projective codes over \mathbb Z4 with type 4^k12^{k2} . [ABSTRACT FROM PUBLISHER]

In this paper, we present a construction of linear codes over \mathbb F2^{t} from Boolean functions, which is a generalization of Ding’s method. Based on this construction, we give two classes of linear codes \tilde \mathcal Cf and \mathcal Cf over \mathbb F2^{t} from a Boolean function f: \mathbb Fq\rightarrow \mathbb F 2 , where q=2^n and \mathbb F2^{t} is some subfield of \mathbb Fq . The complete weight enumerator of \tilde \mathcal Cf can be easily determined from the Walsh spectrum of f , while the weight distribution of the code f . As applications of this construction, we show several series of linear codes over \mathbb F2^t with two or three weights by using bent, semibent, monomial and quadratic Boolean function f . [ABSTRACT FROM PUBLISHER]

Locally repairable codes (LRCs) are a class of codes designed for the local correction of erasures. They have received considerable attention in recent years due to their applications in distributed storage. Most existing results on LRCs do not explicitly take into consideration the field size $q$ , i.e., the size of the code alphabet. In particular, for the binary case, only a few results are known. In this paper, we present an upper bound on the minimum distance $d$ of linear LRCs with availability, based on the work of Cadambe and Mazumdar. The bound takes into account the code length $n$ , dimension $k$ , locality $r$ , availability $t$ , and field size $q$ . Then, we study the binary linear LRCs in three aspects. First, we focus on analyzing the locality of some classical codes, i.e., cyclic codes and Reed–Muller codes, and their modified versions, which are obtained by applying the operations of extend, shorten, expurgate, augment, and lengthen. Next, we construct LRCs using phantom parity-check symbols and multi-level tensor product structure, respectively. Compared with other previous constructions of binary LRCs with fixed locality or minimum distance, our construction is much more flexible in terms of code parameters, and gives various families of high-rate LRCs, some of which are shown to be optimal with respect to their minimum distance. Finally, the availability of LRCs is studied. We investigate the locality and availability properties of several classes of one-step majority-logic decodable codes, including cyclic simplex codes, cyclic difference-set codes, and 4-cycle free regular low-density parity-check codes. We also show the construction of a long LRC with availability from a short one-step majority-logic decodable code. [ABSTRACT FROM PUBLISHER]

Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems, and communication systems as they have efficient encoding and decoding algorithms. Perfect nonlinear monomials were employed to construct optimal ternary cyclic codes with parameters [3^m-1, 3^m-1-2m, 4] by Carlet, Ding, and Yuan in 2005. In this paper, almost perfect nonlinear monomials, and a number of other monomials over GF(3^m) are used to construct optimal ternary cyclic codes with the same parameters. Nine open problems on such codes are also presented. [ABSTRACT FROM AUTHOR]

Cyclic codes with two zeros and their dual codes have been a subject of study for many years. However, their weight distributions are known only for a few cases. In this paper, the weight distributions of the duals of the cyclic codes with two zeros are settled for a few more cases. [ABSTRACT FROM AUTHOR]

Irreducible cyclic codes have been an interesting subject of study for a long time. Their weight distribution is known in only a few cases. In this paper, the weight distribution of the irreducible cyclic codes in a number of other cases is determined. The number of nonzero weights in the codes dealt with in this paper varies between one and four. [ABSTRACT FROM AUTHOR]

Algebraic network information theory is an emerging facet of network information theory, studying the achievable rates of random code ensembles that have algebraic structure, such as random linear codes. A distinguishing feature is that linear combinations of codewords can sometimes be decoded more efficiently than codewords themselves. The present work further develops this framework by studying the simultaneous decoding of multiple messages. Specifically, consider a receiver in a multi-user network that wishes to decode several messages. Simultaneous joint typicality decoding is one of the most powerful techniques for determining the fundamental limits at which reliable decoding is possible. This technique has historically been used in conjunction with random i.i.d. codebooks to establish achievable rate regions for networks. Recently, it has been shown that, in certain scenarios, nested linear codebooks in conjunction with “single-user” or sequential decoding can yield better achievable rates. For instance, the compute–forward problem examines the scenario of recovering $L \le K$ linear combinations of transmitted codewords over a $K$ -user multiple-access channel (MAC), and it is well established that linear codebooks can yield higher rates. This paper develops bounds for simultaneous joint typicality decoding used in conjunction with nested linear codebooks, and applies them to obtain a larger achievable region for compute–forward over a $K$ -user discrete memoryless MAC. The key technical challenge is that competing codeword tuples that are linearly dependent on the true codeword tuple introduce statistical dependencies, which requires careful partitioning of the associated error events. [ABSTRACT FROM AUTHOR]

An $[{n}, {k}, {n}-{k}+1]$ linear code is called an MDS code. An $[{n}, {k}, {n}-{k}]$ linear code is said to be almost maximum distance separable (almost MDS or AMDS for short). A code is said to be near maximum distance separable (near MDS or NMDS for short) if the code and its dual code both are almost maximum distance separable. The first near MDS code was the [11, 6, 5] ternary Golay code discovered in 1949 by Golay. This ternary code holds 4-designs, and its extended code holds a Steiner system ${S}(5, 6, 12)$ with the largest strength known. In the past 70 years, sporadic near MDS codes holding t-designs were discovered and a lot of infinite families of near MDS codes over finite fields were constructed. However, the question as to whether there is an infinite family of near MDS codes holding an infinite family of t-designs for $ {t}\geq 2$ remains open for 70 years. This paper settles this long-standing problem by presenting an infinite family of near MDS codes over $ {GF}(3^{ {s}})$ holding an infinite family of 3-designs and an infinite family of near MDS codes over $ {GF}(2^{2 {s}})$ holding an infinite family of 2-designs. The subfield subcodes of these two families of codes are also studied, and are shown to be dimension-optimal or distance-optimal. [ABSTRACT FROM AUTHOR]

Limited magnitude asymmetric error model is well suited for flash memory. In this paper, we consider the construction of asymmetric codes correcting single error over $\mathbb {Z}_{2^{k}r}$ which is based on so called $B_{1}[{4}](2^{k}r)$ set. In fact, we reduce the construction of a maximal size $B_{1}[{4}](2^{k}r)$ set for $k\geq 3$ to the construction of a maximal size $B_{1}[{4}](2^{k-3}r)$ set. Finally, we give an explicit formula of a maximal size $B_{1}[{4}](4r)$ set and some lower bounds of a maximal size $B_{1}[{4}](2r)$ set. By computer searching up to $q\leq 106$ , we conjecture that those lower bounds are tight. [ABSTRACT FROM AUTHOR]

The binary Hamming codes with parameters $[{2^{m}-1, 2^{m}-1-m, 3}]$ are perfect. Their extended codes have parameters $[{2^{m}, 2^{m}-1-m, 4}]$ and are distance-optimal. The first objective of this paper is to construct a class of binary linear codes with parameters $[{2^{m+s}+2^{s}-2^{m},2^{m+s}+2^{s}-2^{m}-2m-2,4}]$ , which have better information rates than the class of extended binary Hamming codes, and are also distance-optimal. The second objective is to construct a class of distance-optimal binary codes with parameters $[{2^{m}+2, 2^{m}-2m, 6}]$. Both classes of binary linear codes have new parameters. [ABSTRACT FROM AUTHOR]

In this paper, locally repairable codes which have optimal minimum Hamming distance with respect to the bound presented by Prakash et al. are considered. New upper bounds on the length of such optimal codes are derived. The new bounds apply to more general cases, and have weaker requirements compared with the known ones. In this sense, they both improve and generalize previously known bounds. Further, optimal codes are constructed, whose length is order-optimal with respect to the new upper bounds. Notably, the length of the codes is super-linear in the alphabet size. [ABSTRACT FROM AUTHOR]

In this paper we consider self-dual NRT codes, that is, self-dual codes in the metric space endowed with the Niederreiter-Rosenbloom-Tsfasman metric (NRT metric) and their shape enumerators as defined by Barg and Park. We use polynomial invariant theory to describe the shape enumerator of a binary self-dual NRT code, even self-dual NRT code, and weak doubly even self-dual NRT code in $ {M}_{ {n},2}(\mathbb {F}_{2})$. Motivated by these results, we describe the number of invariant polynomials that we must find to describe the shape enumerator of a self-dual NRT code in $ {M}_{ {n}, {s}}(\mathbb {F}_{2})$. We define the ordered flip of a matrix $ {A}\in {M}_{ {k},{ { ns}}}(\mathbb {F}_{ {q}})$ and present some constructions of self-dual NRT codes over $\mathbb {F}_{ {q}}$. We further give an application of ordered flip to the classification of self-dual NRT codes of dimension two. [ABSTRACT FROM AUTHOR]

MDS codes and self-dual codes are important families of classical codes in coding theory. Therefore, it is of interest to investigate MDS self-dual codes. The existence of MDS self-dual codes over finite field ${\mathbb F}_{q}$ is competely solved for $q$ is even. In the literature, there are many known constructions of MDS self-dual codes for $q$ is odd. In this paper, we present a unified approach on the existence of MDS self-dual codes with concise statements and simplified proof. It is illustrated that some of known results can also be stated in this framework. Furthermore, we can obtain some new MDS self-dual codes, especially for the case when $q$ is not a square. [ABSTRACT FROM AUTHOR]

In this paper, we study self-dual codes over finite fields using tools from algebraic function fields in one variable. An algebraic geometry code of length $n$ is defined using two divisors $G$ and $D=P_{1}+\cdots +P_{n}$. We characterize self-orthogonality of the genus zero code in terms of the divisors $G$ , $D$ and the value of a well-chosen derivative polynomial at points $(P_{i})_{1\le i\le n}$. We explore the existence problem of MDS self-dual codes in the odd characteristic case, and we explicitly construct families of new MDS self-dual codes. [ABSTRACT FROM AUTHOR]

Hermitian functional and differential codes are AG-codes defined on a Hermitian curve. To ensure good performance, the divisors defining such AG-codes have to be carefully chosen, exploiting the rich combinatorial and algebraic properties of the Hermitian curves. In this paper, the case of differential codes $C_{\Omega }(\mathtt {D},m\mathtt {T})$ on the Hermitian curve $\mathscr {H}_{q^{3}}$ defined over $\mathbb {F}_{q^{6}}$ is worked out where $\mathop {\mathrm {supp}}\nolimits (\mathtt {T}):= \mathscr {H}_{q^{3}}(\mathbb {F}_{q^{2}})$ , the set of all $\mathbb {F}_{q^{2}}$ -rational points of $\mathscr {H}_{q^{3}}$ , while $\mathtt {D}$ is taken, as usual, to be the sum of the points in the complementary set $D = \mathscr {H}_{q^{3}}(\mathbb {F}_{q^{6}})\setminus \mathscr {H} _{q^{3}}(\mathbb {F}_{q^{2}})$. For certain values of $m$ , such codes $C_{\Omega }(\mathtt {D},m\mathtt {T})$ have better minimum distance compared with true values of 1-point Hermitian codes. The automorphism group of $C_{L}(\mathtt {D},m\mathtt {T})$ , $m\leq q^{3}-2$ , is isomorphic to $PGU(3,q)$. [ABSTRACT FROM AUTHOR]

Gabidulin codes are the first general construction of linear codes that are maximum rank distant (MRD). They have found applications in linear network coding, for example, when the transmitter and receiver are oblivious to the inner workings and topology of the network (the so-called incoherent regime). The reason is that Gabidulin codes can be used to map information to linear subspaces, which in the absence of errors cannot be altered by linear operations, and in the presence of errors can be corrected if the subspace is perturbed by a small rank. Furthermore, in distributed coding and distributed systems, one is led to the design of error correcting codes whose generator matrix must satisfy a given support constraint. In this paper, we give necessary and sufficient conditions on the support of the generator matrix that guarantees the existence of Gabidulin codes and general MRD codes. When the rate of the code is not very high, this is achieved with the same field size necessary for Gabidulin codes with no support constraint. When these conditions are not satisfied, we characterize the largest possible rank distance under the support constraints and show that they can be achieved by subcodes of Gabidulin codes. The necessary and sufficient conditions are identical to those that appear for MDS codes which were recently proven by Yildiz et al. and Lovett in the context of settling the GM-MDS conjecture. [ABSTRACT FROM AUTHOR]

In this paper, we construct several classes of maximum distance separable (MDS) codes via generalized Reed-Solomon (GRS) codes and extended GRS codes, where we can determine the dimensions of their Euclidean hulls or Hermitian hulls. It turns out that the dimensions of Euclidean hulls or Hermitian hulls of the codes in our constructions can take all or almost all possible values. As a consequence, we can apply our results to entanglement-assisted quantum error-correcting codes (EAQECCs) and obtain several new families of MDS EAQECCs with flexible parameters. The required number of maximally entangled states of these MDS EAQECCs can take all or almost all possible values. Moreover, several new classes of q-ary MDS EAQECCs of length $ {n} > {q}+1$ are also obtained. [ABSTRACT FROM AUTHOR]