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In this paper, we study the decoding of dual-containing codes from Hermitian tower and applications to quantum codes. The contribution of this paper is threefold. First, we construct the quantum stabilizer codes from the Hermitian tower. Second, we provide a deterministic decoding algorithm with decoding radius that almost achieves the optimal decoding radius, i.e., $(1-R)/4$ , where $R$ is the rate. Last and most importantly, we present a Monte Carlo algorithm with decoding radius roughly equal to $(1-R)/3$ , which is beyond the optimal decoding radius $(1-R)/4$ . There are several features in this paper. First of all, we employ a differential for the Hermitian tower. This differential plays a crucial role for decoding. We also extend our decoding by passing to the constant field extension. This constant field extension makes the decoding work perfectly. [ABSTRACT FROM PUBLISHER]

In this paper, we introduce a model of a distributed storage system that is locally recoverable from any single server failure. Unlike the usual local recovery model of codes for distributed storage, this model accounts for the fact that each server or storage node in a network is connectable to only some, and not all other, nodes. This may happen for reasons such as physical separation, inhomogeneity in storage platforms, and so forth. We estimate the storage capacity of both undirected and directed networks under this model and propose some constructive schemes. From a coding theory point of view, we show that this model is approximately dual of the well-studied index coding problem. Furthermore, in this paper, we extend the above model to handle multiple server failures. Among other results, we provide an upper bound on the minimum pairwise distance of a set of words that can be stored in a graph with the local repair guarantee. The well-known impossibility bounds on the distance of locally recoverable codes follow from our result. [ABSTRACT FROM PUBLISHER]

The locally repairable code (LRC) studied in this paper is an [n,k] linear code of which the value at each coordinate can be recovered by a linear combination of at most r other coordinates. The central problem in this paper is to determine the largest possible minimum distance for LRCs. First, an integer programming-based upper bound is derived for any LRC. Then, by solving the programming problem under certain conditions, an explicit upper bound is obtained for LRCs with parameters n1>n2 , where n1 = \lceil ({n}/{r+1}) \rceil and n2 = n1 (r+1) - n . Finally, an explicit construction for LRCs attaining this upper bound is presented over the finite field \mathbb F2^{m} , where m\geq n1r . Based on these results, the largest possible minimum distance for all LRCs with r \le \sqrt n-1 has been definitely determined, which is of great significance in practical use. [ABSTRACT FROM PUBLISHER]

Because of their efficient encoding and decoding algorithms, cyclic codes—an interesting class of linear codes—are widely used in communication systems, storage devices, and consumer electronics. BCH codes form a special class of cyclic codes, and are usually among the best cyclic codes. A subclass of good BCH codes is the narrow-sense primitive BCH codes. However, the dimension and minimum distance of these codes are not known in general. The main objective of this paper is to study the dimension and minimum distances of a subclass of the narrow-sense primitive BCH codes with design distance \delta =(q-\ell 0)q^{m-\ell 1-1}-1 for certain pairs (\ell 0, \ell 1) , where 0 \leq \ell 0 \leq q-2 and 0 \leq \ell 1 \leq m-1 . The parameters of other related classes of BCH codes are also investigated, and some open problems are proposed in this paper. [ABSTRACT FROM PUBLISHER]

This paper studies the parameters for which binary Reed–Muller (RM) codes can be decoded successfully on the binary erasure channel and binary symmetry channel, and, in particular, when can they achieve capacity for these two classical channels. Necessarily, this paper also studies the properties of evaluations of multivariate $GF(2)$ polynomials on the random sets of inputs. For erasures, we prove that RM codes achieve capacity both for very high rate and very low rate regimes. For errors, we prove that RM codes achieve capacity for very low rate regimes, and for very high rates, we show that they can uniquely decode at about the square root of the number of errors at capacity. The proofs of these four results are based on different techniques, which we find interesting in their own right. In particular, we study the following questions about $E(m,r)$ , the matrix whose rows are the truth tables of all the monomials of degree $\leq r$ in $m$ variables. What is the most (resp. least) number of random columns in $E(m,r)$ that define a submatrix having full column rank (resp. full row rank) with high probability? We obtain tight bounds for very small (resp. very large) degrees $r$ , which we use to show that RM codes achieve capacity for erasures in these regimes. Our decoding from random errors follows from the following novel reduction. For every linear code $C$ of sufficiently high rate, we construct a new code $C'$ obtained by tensorizing $C$ , such that for every subset $S$ of coordinates, if $C$ can recover from erasures in $S$ , then $C'$ can recover from errors in $S$ . Specializing this to the RM codes and using our results for erasures imply our result on the unique decoding of the RM codes at high rate. Finally, two of our capacity achieving results require tight bounds on the weight distribution of RM codes. We obtain such bounds extending the recent bounds from constant degree to linear degree polynomials. [ABSTRACT FROM PUBLISHER]

New nonasymptotic random coding theorems (with error probability $\epsilon $ and finite block length $n$ ) based on Gallager parity check ensemble and general parity check ensembles are derived in this paper. The resulting nonasymptotic achievability bounds, when combined with nonasymptotic equipartition properties developed in this paper, can be easily computed. Analytically, these nonasymptotic achievability bounds are shown to be asymptotically tight up to the second order of the coding rate as $n$ goes to infinity with either constant or subexponentially decreasing $\epsilon $ in the case of Gallager parity check ensemble, and to imply that low density parity check (LDPC) codes be capacity-achieving in the case of LDPC ensembles. Numerically, they are also compared favorably, for finite $n$ and $\epsilon $ of practical interest, with existing nonasymptotic achievability bounds in the literature. [ABSTRACT FROM AUTHOR]

A classical method of constructing a linear code over \mathrm GF(q) with a $t$ -design is to use the incidence matrix of the $t$ -design as a generator matrix over \mathrm GF(q) of the code. This approach has been extensively investigated in the literature. In this paper, a different method of constructing linear codes using specific classes of 2-designs is studied, and linear codes with a few weights are obtained from almost difference sets, difference sets, and a type of 2-designs associated to semibent functions. Two families of the codes obtained in this paper are optimal. The linear codes presented in this paper have applications in secret sharing and authentication schemes, in addition to their applications in consumer electronics, communication and data storage systems. A coding-theory approach to the characterization of highly nonlinear Boolean functions is presented. [ABSTRACT FROM AUTHOR]

Linear receivers are often used to reduce the implementation complexity of multiple-antenna systems. In a traditional linear receiver architecture, the receive antennas are used to separate out the codewords sent by each transmit antenna, which can then be decoded individually. Although easy to implement, this approach can be highly suboptimal when the channel matrix is near singular. This paper develops a new linear receiver architecture that uses the receive antennas to create an effective channel matrix with integer-valued entries. Rather than attempting to recover transmitted codewords directly, the decoder recovers integer combinations of the codewords according to the entries of the effective channel matrix. The codewords are all generated using the same linear code, which guarantees that these integer combinations are themselves codewords. Provided that the effective channel is full rank, these integer combinations can then be digitally solved for the original codewords. This paper focuses on the special case where there is no coding across transmit antennas and no channel state information at the transmitter(s), which corresponds either to a multiuser uplink scenario or to single-user V-BLAST encoding. In this setting, the proposed integer-forcing linear receiver significantly outperforms conventional linear architectures such as the zero forcing and linear minimum mean-squared error receiver. In the high signal-to-noise ratio regime, the proposed receiver attains the optimal diversity-multiplexing tradeoff for the standard multiple-input multiple-output (MIMO) channel with no coding across transmit antennas. It is further shown that in an extended MIMO model with interference, the integer-forcing linear receiver achieves the optimal generalized degrees of freedom. [ABSTRACT FROM AUTHOR]

This paper considers a base station that delivers packets to multiple receivers through a sequence of coded transmissions. All receivers overhear the same transmissions. Each receiver may already have some of the packets as side information, and requests another subset of the packets. This problem is known as the index coding problem and can be represented by a bipartite digraph. An integer linear program is developed that provides a lower bound on the minimum number of transmissions required for any coding algorithm. Conversely, its linear programming relaxation is shown to provide an upper bound that is achievable by a simple form of vector linear coding. Thus, the information theoretic optimum is bounded by the integrality gap between the integer program and its linear relaxation. In the special case, when the digraph has a planar structure, the integrality gap is shown to be zero, so that exact optimality is achieved. Finally, for nonplanar problems, an enhanced integer program is constructed that provides a smaller integrality gap. The dual of this problem corresponds to a more sophisticated partial clique coding strategy that time-shares between maximum distance separable codes. This paper illuminates the relationship between index coding, duality, and integrality gaps between integer programs and their linear relaxations. [ABSTRACT FROM PUBLISHER]

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]

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]

The projective special linear group ${\mathrm {PSL}}_{2}(n)$ is 2-transitive for all primes $n$ and 3-homogeneous for $n \equiv 3 \pmod {4}$ on the set $\{0,1, \ldots, n-1, \infty \}$. It is known that the extended odd-like quadratic residue codes are invariant under ${\mathrm {PSL}}_{2}(n)$. Hence, the extended quadratic residue codes hold an infinite family of 2-designs for primes $n \equiv 1 \pmod {4}$ , an infinite family of 3-designs for primes $n \equiv 3 \pmod {4}$. To construct more $t$ -designs with $t \in \{2, 3\}$ , one would search for other extended cyclic codes over finite fields that are invariant under the action of ${\mathrm {PSL}}_{2}(n)$. The objective of this paper is to prove that the extended quadratic residue binary codes are the only nontrivial extended binary cyclic codes that are invariant under ${\mathrm {PSL}}_{2}(n)$. [ABSTRACT FROM AUTHOR]

The multipoint codes from algebraic curves are a broad class of algebraic geometry codes derived from algebraic functions, which have multiple poles/zeros on their defining curves. Each of them is defined as either a primal code or a dual code. The dual one-point codes which are viewed as a subclass can be decoded efficiently up to the Feng–Rao bound by using the Berlekamp–Massey–Sakata (BMS) algorithm with majority logic. Since a primal code is equivalent to a dual code, one can decode as either of them, while their decoding methods are different. Recently, we published a fast method for decoding primal multipoint codes from curves based on the vectorial BMS algorithm. But, that is neither for dual codes nor up to the Goppa bound d\mathrm{ Goppa} . Although we can guarantee theoretically that every error vector of weight only up to (1/2)(d\mathrm{ Goppa}-g) can be corrected, where the integer g is the genus of the defining curve, the simulation shows that the method can correct most error patterns of weight up to ({1}/{2})d_{\mathrm{ Goppa}} . In this paper we present a fast method for decoding dual multipoint codes from algebraic curves up to the Kirfel–Pellikaan bound, based on the vectorial BMS algorithm with majority logic, and show that algebraic geometry codes from generic algebraic curves can be decoded up to the Goppa bound efficiently. Similar to the case of one-point codes, the computational complexity of decoding is {\mathcal{ O}}(a_{1}n^{2}) , where the integer a1 is the minimum nonzero pole order of algebraic functions on the defining curve and the integer n is the code length, and in particular, \mathcal O(n^(7/3)) for Hermitian codes. This complexity is less than the complexity \mathcal O(a1gn^{2}) of Lee’s method for decoding dual multipoint codes as a unique alternative. [ABSTRACT FROM PUBLISHER]

Secret-sharing schemes with optimal and universal communication overheads have been obtained independently by Bitar et al. and Huang et al. However, their constructions require a finite field of size $q > n$ , where $n$ is the number of shares, and do not provide strong security. In this paper, we give a general framework to construct communication efficient secret-sharing schemes based on sequences of nested linear codes, which allows to use, in particular, algebraic geometry codes and allows to obtain strongly secure and communication-efficient schemes. Using this framework, we obtain: 1) schemes with universal and close to optimal communication overheads for arbitrarily large lengths $n$ and a fixed finite field; 2) the first construction of schemes with universal and optimal communication overheads and optimal strong security (for restricted lengths), having, in particular, the component-wise security advantages of perfect schemes and the security and storage efficiency of ramp schemes; and 3) schemes with universal and close to optimal communication overheads and close to optimal strong security defined for arbitrarily large lengths $n$ and a fixed finite field. [ABSTRACT FROM PUBLISHER]

Universal security over a network with linear network coding has been intensively studied. However, previous linear codes and code pairs used for this purpose were linear over a larger field than that used on the network, which restricts the possible packet lengths of optimal universal secure codes, does not allow to apply known list-decodable rank-metric codes and requires performing operations over a large field. In this paper, we introduce new parameters (relative generalized matrix weights and relative dimension/rank support profile) for code pairs that are linear over the field used in the network, and show that they measure the universal security performance of these code pairs. For one code and non-square matrices, generalized matrix weights coincide with the existing Delsarte generalized weights, hence we prove the connection between these latter weights and secure network coding, which was left open. As main applications, the proposed new parameters enable us to: 1) obtain optimal universal secure linear codes on noiseless networks for all possible packet lengths, in particular for packet lengths not considered before, 2) obtain the first universal secure list-decodable rank-metric code pairs with polynomial-sized lists, based on a recent construction by Guruswami et al; and 3) obtain new characterizations of security equivalences of linear codes. Finally, we show that our parameters extend relative generalized Hamming weights and relative dimension/length profile, respectively, and relative generalized rank weights and relative dimension/intersection profile, respectively. [ABSTRACT FROM AUTHOR]

A code is said to be an $r$ -local locally repairable code (LRC) if each of its coordinates can be repaired by accessing at most r$ other coordinates. When some of the r$ coordinates are also erased, the r$ -local LRC cannot accomplish the local repair, which leads to the concept of (r,\delta)$ -locality. A q is said to have (r, \delta)$ -locality ( \delta \ge 2$ ) if for each coordinate i with support containing $i$ , whose length is at most $r + \delta - 1$ , and whose minimum distance is at least $\delta$ . The $(r, \delta)$ -LRC can tolerate $\delta -1$ erasures in every local code (i.e., punctured subcode), which degenerates to an $r$ -local LRC when $\delta =2$ . A $q$ -ary $(r,\delta)$ LRC is called optimal if it meets the singleton-like bound for $(r,\delta)$ -LRCs. A class of optimal $q$ -ary cyclic $r$ -local LRCs with lengths $n\mid q-1$ were constructed by Tamo, Barg, Goparaju, and Calderbank based on the $q$ -ary Reed-Solomon codes. In this paper, we construct a class of optimal $q$ -ary cyclic $(r,\delta)$ -LRCs ( $\delta \ge 2$ ) with length $n\mid q-1$ , which generalizes the results of Tamo et al. Moreover, we construct a new class of optimal $q$ -ary cyclic $r$ -local LRCs with lengths $n\mid q+1$ and a new class of optimal $q$ -ary cyclic $(r,\delta)$ -LRCs ( $\delta \ge 2$ ) with lengths $n\mid q+1$ . The constructed optimal LRCs with length $n=q+1$ have the best-known length for a given finite field with size $q$ when the minimum distance is larger than 4. [ABSTRACT FROM AUTHOR]

In this paper, we construct a new family of quantum synchronizable codes from repeated-root cyclic codes of lengths p^s and lp^s over \mathbb Fq , where $s\geq 1~and l\geq 2$ are integers, and $p\geq 3$ is the odd characteristic. Within some loose limitations, these synchronizable codes can possess the best possible capability in synchronization recovery, and therefore, enriches the variety of good quantum synchronizable codes. Furthermore, by using known techniques in classical coding theory which convert the computation of the minimum distance of a repeated-root cyclic code to that of a shorter simple-root cyclic code, we prove that the repeated-root cyclic codes of lengths p^s and lp^s are in general better than narrow-sense BCH codes of close lengths in terms of minimum distances, and thereby enable the obtained synchronizable codes to correct more Pauli errors. [ABSTRACT FROM PUBLISHER]

We study the communication efficient secret sharing (CESS) problem. A classical threshold secret sharing scheme encodes a secret into $n$ shares given to n$ parties, such that any set of at least t$ , t n$ . However, Staircase codes may require dividing the secret and the shares into many symbols. We also describe how Staircase codes can be used to construct threshold changeable secret sharing with minimum storage cost, i.e., minimum share size. [ABSTRACT FROM AUTHOR]

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]

Symbol-pair codes introduced by Cassuto and Blaum (2010) are designed to protect against pair errors in symbol-pair read channels. The higher the minimum pair distance, the more pair errors the code can correct. Maximum distance separable (MDS) symbol-pair codes are optimal in the sense that pair distance cannot be improved for given length and code size. The contribution of this paper is twofold. First, we present three lower bounds for the minimum pair distance of constacyclic codes, the first two of which generalize the previously known results due to Cassuto and Blaum (2011) and Kai et al. (2015). The third one exhibits a lower bound for the minimum pair distance of repeated-root cyclic codes. Second, we obtain new MDS symbol-pair codes with minimum pair distance seven and eight through repeated-root cyclic codes. [ABSTRACT FROM AUTHOR]

Index coding models broadcast networks in which a sender sends different messages to different receivers simultaneously, where each receiver may know some of the messages a priori. The aim is to find the minimum (normalized) index codelength that the sender sends. This paper considers unicast index coding, where each receiver requests exactly one message, and each message is requested by exactly one receiver. Each unicast index-coding instances can be fully described by a directed graph and vice versa, where each vertex corresponds to one receiver. For any directed graph representing a unicast index-coding instance, we show that if a maximum acyclic induced subgraph (MAIS) is obtained by removing two or fewer vertices from the graph, then the minimum index codelength equals the number of vertices in the MAIS, and linear codes are optimal for the corresponding index-coding instance. Using this result, we solved all unicast index-coding instances with up to five receivers, which correspond to all graphs with up to five vertices. For 9818 non-isomorphic graphs among all graphs up to five vertices, we obtained the minimum index codelength for all message alphabet sizes; for the remaining 28 graphs, we obtained the minimum index codelength if the message alphabet size is k^2 for any positive integer $k$ . This work complements the result by Arbabjolfaei et al. (ISIT 2013), who solved all unicast index-coding instances with up to five receivers in the asymptotic regime, where the message alphabet size tends to infinity. [ABSTRACT FROM PUBLISHER]

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]

Regenerating codes and codes with locality are two coding schemes that have recently been proposed, which in addition to ensuring data collection and reliability, also enable efficient node repair. In a situation where one is attempting to repair a failed node, regenerating codes seek to minimize the amount of data downloaded for node repair, while codes with locality attempt to minimize the number of helper nodes accessed. This paper presents results in two directions. In one, this paper extends the notion of codes with locality so as to permit local recovery of an erased code symbol even in the presence of multiple erasures, by employing local codes having minimum distance >2. An upper bound on the minimum distance of such codes is presented and codes that are optimal with respect to this bound are constructed. The second direction seeks to build codes that combine the advantages of both codes with locality as well as regenerating codes. These codes, termed here as codes with local regeneration, are codes with locality over a vector alphabet, in which the local codes themselves are regenerating codes. We derive an upper bound on the minimum distance of vector-alphabet codes with locality for the case when their constituent local codes have a certain uniform rank accumulation property. This property is possessed by both minimum storage regeneration (MSR) and minimum bandwidth regeneration (MBR) codes. We provide several constructions of codes with local regeneration which achieve this bound, where the local codes are either MSR or MBR codes. Also included in this paper, is an upper bound on the minimum distance of a general vector code with locality as well as the performance comparison of various code constructions of fixed block length and minimum distance. [ABSTRACT FROM PUBLISHER]

Following the very recent studies on \mathbb Z2\mathbb Z4 -additive codes, \mathbb Z2\mathbb Z2[u] -linear codes have been introduced by Aydogdu et al. In this paper, we introduce and study the algebraic structure of cyclic, constacyclic codes and their duals over the R -module \mathbb {Z}_{2}^\alpha R^\beta where R=\mathbb {Z}_{2}+u\mathbb {Z}_{2}=\left \{{0,1,u,u+1}\right \} is the ring with four elements and u^2=0 . We determine the generating independent sets and the types and sizes of both such codes and their duals. Finally, we present a bound and an optimal family of codes attaining this bound and also give some illustrative examples of binary codes that have good parameters which are obtained from the cyclic codes in \mathbb Z_2^\alpha R^\beta . [ABSTRACT FROM AUTHOR]

In this paper, we consider metrics determined by hierarchical posets and give explicit formulas for the main parameters of a linear code: the minimum distance and the packing, covering, and Chebyshev radii of a code. We also present ten characterizations of hierarchical poset metrics, including new characterizations and simple new proofs to the known ones. [ABSTRACT FROM AUTHOR]

A normal rational curve of the (k-1) -dimensional projective space over {\mathbb F}_{q} is an arc of size q+1$ , since any k$ points of the curve span the whole space. In this paper, we will prove that if q$ is odd, then a subset of size 3k-6$ of a normal rational curve cannot be extended to an arc of size q+2$ . In fact, we prove something slightly stronger. Suppose that q$ is odd and E$ is a (2k-3)$ -subset of an arc G$ of size 3k-6$ . If G$ projects to a subset of a conic from every (k-3)$ -subset of E of odd characteristic, which can be extended to a Reed–Solomon code of length $q+1$ , cannot be extended to a linear maximum distance separable code of length $q+2$ . [ABSTRACT FROM AUTHOR]

We consider a graphical approach to index coding. As cycles have been shown to provide coding gain, cycles and cliques (a specific type of overlapping cycles) have been exploited in an existing literature. In this paper, we define a more general form of overlapping cycles, called the interlinked-cycle ( \mathsf IC ) structure, that generalizes cycles and cliques. We propose a scheme, called the interlinked-cycle-cover ( \mathsf ICC ) scheme, that leverages \mathsf IC structures in digraphs to construct scalar linear index codes. We characterize a class of infinitely many digraphs where our proposed scheme is optimal over all linear and non-linear index codes. Consequently, for this class of digraphs, we indirectly prove that scalar linear index codes are optimal. Furthermore, we show that the \mathsf ICC scheme can outperform all the existing graph-based schemes (including partial-clique-cover and fractional-local-chromatic number schemes), and a random-coding scheme (namely, composite coding) for certain graphs. [ABSTRACT FROM AUTHOR]

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]

A linear complementary dual (LCD) code is a linear code with complimentary dual. LCD codes have been extensively studied in literature. On the other hand, maximum distance separable (MDS) codes are an important class of linear codes that have found wide applications in both theory and practice. However, little is known about MDS codes with complimentary duals. The main purpose of this paper is to construct several classes of MDS codes with complimentary duals, i.e., LCD MDS codes, through generalized Reed-Solomon codes. [ABSTRACT FROM PUBLISHER]

Reed–Muller (RM) codes encode an m -variate polynomial of degree at most r by evaluating it on all points in \0,1\^m . We denote this code by RM(r,m) . The minimum distance of RM(r,m) is 2^{m-r} and so it cannot correct more than half that number of errors in the worst case. For random errors one may hope for a better result. In this paper we give an efficient algorithm (in the block length n=2^{m} ) for decoding random errors in RM codes far beyond the minimum distance. Specifically, for low-rate codes (of degree r=o(\sqrt m) ), we can correct a random set of (1/2-o(1))n errors with high probability. For high rate codes (of degree m-r for r=o(\sqrt {m/\log m}) ), we can correct roughly m^{r/2}$ errors. More generally, for any integer $r$ , our algorithm can correct any error pattern in $RM(m-(2r+2),m)$ , for which the same erasure pattern can be corrected in $RM(m-(r+1),m)$ . The results above are obtained by applying recent results of Abbe, Shpilka, and Wigderson (STOC, 2015) and Kudekar et al. (STOC, 2016) regarding the ability of RM codes to correct random erasures. The algorithm is based on solving a carefully defined set of linear equations and thus it is significantly different than other algorithms for decoding RM codes that are based on the recursive structure of the code. It can be seen as a more explicit proof of a result of Abbe et al. that shows a reduction from correcting erasures to correcting errors, and it also bares some similarities with the error-locating pair method of Pellikaan, Duursma, and Kötter that generalizes the Berlekamp–Welch algorithm for decoding Reed–Solomon codes. [ABSTRACT FROM PUBLISHER]

In this paper, we prove that Schubert union codes are Tanner codes constructed from the point–line incidence geometry inherited from the Grassmannian. Our proof is based on an iterative encoding algorithm for Tanner codes. This encoder determines the entries of a code word of a Tanner code from the entries in a given subset of its positions. As a result, we find sufficient conditions on the initial positions such that a code word is determined from the component codes only. This algorithm has linear complexity in the code length. We also use this encoder to determine the minimum distance of Schubert union codes in terms of the minimum distance of the Schubert varieties contained therein. [ABSTRACT FROM PUBLISHER]

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]

In this paper, we obtain an improved lower bound on the error exponent of the memoryless multiple-access channel via the use of linear codes, thus demonstrating that structure can be beneficial even when capacity may be achieved via random codes. We show that if the multiple-access channel is additive over a finite field, then any error probability, and hence any error exponent, achievable by a linear code for the associated single-user channel, is also achievable for the multiple-access channel. In particular, linear codes allow to attain joint expurgation, and hence, attain the single-user expurgated exponent of the single-user channel, whenever the latter is achieved by a uniform distribution. Thus, for additive channels, at low rates, where expurgation is needed, our approach strictly improves performance over previous results, where expurgation was used for at most one of the users. Even when the multiple-access channel is not additive, it may be transformed into such a channel. While the transformation is information-lossy, we show that the distributed structure gain in some “nearly additive” cases outweighs the loss. Finally, we apply a similar approach to the Gaussian multiple-access channel. While we believe that due to the power constraints, it is impossible to attain the single-user error exponent, we do obtain an improvement over the best known achievable error exponent, given by Gallager, for certain parameters. This is accomplished using a nested lattice triplet with judiciously chosen parameters. [ABSTRACT FROM PUBLISHER]

Golay complementary sequences and complementary sets have been proposed to deal with the high peak-to-average power ratio (PAPR) problem in orthogonal frequency division multiplexing (OFDM) system. The existing constructions of complementary sets based on generalized Boolean functions are limited to lengths, which are powers of two. In this paper, we propose novel constructions of binary and nonbinary complementary sets of non-power-of-two length. Regardless of whether or not the length of the complementary set is a power of two, its PAPR is still upper bounded by the size of the complementary set. Therefore, the constructed complementary sets can be applied in practical OFDM systems where the number of used subcarriers is not a power of two. In addition, while the binary Golay complementary pairs exist only for limited lengths, the constructed binary complementary sets of size 4 exist for more lengths with PAPR at most 4. [ABSTRACT FROM PUBLISHER]

One central theme in quantum error-correction is to construct quantum codes that have a relatively large minimum distance. In this paper, we first present a construction of classical linear codes based on certain classes of polynomials. Through these classical linear codes, we are able to obtain some new quantum codes. It turns out that some of the quantum codes exhibited here have better parameters than the ones available in the literature. [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]

Combinatorial structures called stopping sets are useful in analyzing the performance of a linear code when coupled with an iterative decoding algorithm over an erasure channel. In this paper, we consider stopping sets of Hermitian codes. [ABSTRACT FROM PUBLISHER]

This paper provides a simplified proof for the existence of nested lattice codebooks allowing to achieve the capacity of the additive white Gaussian noise channel, as well as the optimal rate-distortion tradeoff for a Gaussian source. The proof is self-contained and relies only on basic probabilistic and geometrical arguments. An ensemble of nested lattices that is different, and more elementary, than the one used in the previous proofs is introduced. This ensemble is based on lifting different subcodes of a linear code to the Euclidean space using Construction A. In addition to being simpler, the analysis is less sensitive to the assumption that the additive noise is Gaussian. In particular, for additive ergodic noise channels, it is shown that the achievable rates of the nested lattice coding scheme depend on the noise distribution only via its power. Similarly, the nested lattice source coding scheme attains the same rate-distortion tradeoff for all ergodic sources with the same second moment. [ABSTRACT FROM PUBLISHER]

Cyclic codes with a few weights are very useful in the design of frequency hopping sequences and the development of secret sharing schemes. In this paper, we mainly use Gauss sums to represent the Hamming weights of cyclic codes whose duals have two zeroes. A lower bound of the minimum Hamming distance is determined. In some cases, we give the Hamming weight distributions of the cyclic codes. In particular, we obtain a class of three-weight optimal cyclic codes achieving the Griesmer bound, which generalizes a Vega’s result, and several classes of cyclic codes with a few weights, which solve an open problem proposed by Vega. [ABSTRACT FROM AUTHOR]

The generalized Hamming weights (GHWs) of linear codes are fundamental parameters, the knowledge of which is of great interest in many applications. However, to determine the GHWs of linear codes is difficult in general. In this paper, we study the GHWs for a family of reducible cyclic codes and obtain the complete weight hierarchy in several cases. This is achieved by extending the idea of Yang et al. into higher dimension and by employing some interesting combinatorial arguments. It shall be noted that these cyclic codes may have arbitrary number of nonzeros. [ABSTRACT FROM PUBLISHER]

Rank weights and generalized rank weights have been proved to characterize error and erasure correction, and information leakage in linear network coding, in the same way as Hamming weights and generalized Hamming weights describe classical error and erasure correction, and information leakage in wire-tap channels of type II and code-based secret sharing. Although many similarities between both the cases have been established and proved in the literature, many other known results in the Hamming case, such as bounds or characterizations of weight-preserving maps, have not been translated to the rank case yet, or in some cases have been proved after developing a different machinery. The aim of this paper is to further relate both weights and generalized weights, show that the results and proofs in both cases are usually essentially the same, and see the significance of these similarities in network coding. Some of the new results in the rank case also have new consequences in the Hamming case. [ABSTRACT FROM PUBLISHER]

Most bounds on the size of codes hold for any code, whether linear or not. Notably, the Griesmer bound holds only in the linear case and so optimal linear codes are not necessarily optimal codes. In this paper, we identify code parameters $(q,d,k)$ , namely, field size, minimum distance, and combinatorial dimension, for which the Griesmer bound also holds in the (systematic) nonlinear case. Moreover, we show that the Griesmer bound does not necessarily hold for a systematic code by explicit construction of a family of optimal systematic binary codes. On the other hand, we are able to provide some versions of the Griesmer bound holding for all the systematic codes. [ABSTRACT FROM AUTHOR]

This paper introduces the covering dimension of a linear code over a finite field, which is analogous to the critical exponent of a representable matroid and, thus, generalizes invariants that lie at the heart of several fundamental problems in a coding theory. An upper bound on the covering dimension is proved, improving Kung’s classical bound for the critical exponent. In addition, a construction is given for linear codes that attain equality in this covering dimension bound. [ABSTRACT FROM AUTHOR]

Constacyclic codes which generalize the classical cyclic codes have played important roles in recent constructions of many new quantum maximum distance separable (MDS) codes. However, the mathematical mechanism may not have been fully understood. In this paper, we use pseudo-cyclic codes, which is a further generalization of constacyclic codes, to construct the quantum MDS codes. We can not only provide a unified explanation of many previous constructions, but also produce some new quantum MDS codes. [ABSTRACT FROM AUTHOR]

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

Burst errors are a type of distortion in many data communications and data storage channels. In this paper, we consider the list decodability of codes for single burst error case and phased-burst error case independently. Firstly, we analyze the list decodability of random codes, and we show that the burst list decoding radius and the rate of random codes achieve the Singleton bound and the Gilbert-Varshamov bound for single case and phased case, respectively. Second, we illustrate that cyclic codes and algebraic geometry codes are good burst list-decodable codes for single case and phased case, respectively. [ABSTRACT FROM AUTHOR]

In 1973, Gallager proved that the random-coding bound is exponentially tight for the random code ensemble at all rates, even below expurgation. This result explained that the random-coding exponent does not achieve the expurgation exponent due to the properties of the random ensemble, irrespective of the utilized bounding technique. It has been conjectured that this same behavior holds true for a random ensemble of linear codes. This conjecture is proved in this paper. In addition, it is shown that this property extends to Poltyrev’s random-coding exponent for a random ensemble of lattices. [ABSTRACT FROM PUBLISHER]