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Linear codes with certain special properties have received renewed attention in recent years due to their practical applications. Among them, binary linear complementary dual (LCD) codes play an important role in implementations against side-channel attacks and fault injection attacks. Self-orthogonal codes can be used to construct quantum codes. In this paper, four classes of binary linear codes are constructed via a generic construction which has been intensively investigated in the past decade. Simple characterizations of these linear codes to be LCD or self-orthogonal are presented. Resultantly, infinite families of binary LCD codes and self-orthogonal codes are obtained. Infinite families of binary LCD codes from the duals of these four classes of linear codes are produced. Many LCD codes and self-orthogonal codes obtained in this paper are optimal or almost optimal in the sense that they meet certain bounds on general linear codes. In addition, the weight distributions of two sub-families of the proposed linear codes are established in terms of Krawtchouk polynomials. [ABSTRACT FROM AUTHOR]

In this paper we extend the notion of locally repairable codes to secret sharing schemes. The main problem that we consider is to find optimal ways to distribute the shares of a secret among a set of storage-nodes (participants) such that the content of each node (share) can be recovered by using the contents of only few other nodes, and at the same time the secret can be reconstructed by only some allowable subsets of nodes. As a special case, an eavesdropper observing some set of specific nodes (such as less than certain number of nodes) does not get any information. In other words, we propose to study a locally repairable distributed storage system that is secure against a passive eavesdropper that can observe some subsets of nodes. We provide a number of results related to such systems, including upper bounds and achievability results on the number of bits that can be securely stored with these constraints. In particular, we provide conditions under which a locally repairable code can be turned into a secret sharing scheme and extend the results of secure repairable storage to cooperative repair and storage on networks. In addition, we consider perfect secret sharing schemes over general access structures under locality constraints and give an example of a perfect secret sharing scheme that can have small locality. Finally, we provide a lower bound on the size of a share compared with the size of the secret that shows how locality affects the sizes of shares in a perfect scheme. [ABSTRACT FROM PUBLISHER]

We consider list decoding in the zero-rate regime for two cases—the binary alphabet and the spherical codes in Euclidean space. Specifically, we study the maximal $\tau \in [{0,1}]$ for which there exists an arrangement of $M$ balls of relative Hamming radius $\tau $ in the binary hypercube (of arbitrary dimension) with the property that no point of the latter is covered by $L$ or more of them. As $M\to \infty $ the maximal $\tau $ decreases to a well-known critical value $\tau _{L}$. In this paper, we prove several results on the rate of this convergence. For the binary case, we show that the rate is $\Theta (M^{-1})$ when $L$ is even, thus extending the classical results of Plotkin and Levenshtein for $L=2$. For $L=3$ , the rate is shown to be $\Theta (M^{-({2}/{3})})$. For the similar question about spherical codes, we prove the rate is $\Omega (M^{-1})$ and $O(M^{-({2L}/{L^{2}-L+2})})$. [ABSTRACT FROM AUTHOR]

Like classical block codes, a locally repairable code also obeys the Singleton-type bound (we call a locally repairable code optimal if it achieves the Singleton-type bound). In the breakthrough work of Tamo and Barg, several classes of optimal locally repairable codes were constructed via subcodes of Reed–Solomon codes. Thus, the lengths of the codes given by Tamo and Barg are upper bounded by the code alphabet size $q$. Recently, it was proved through the extension of construction by Tamo and Barg that the length of $q$ -ary optimal locally repairable codes can be $q+1$ by Jin et al. Surprisingly, Barg et al. presented a few examples of $q$ -ary optimal locally repairable codes of small distance and locality with code length achieving roughly $q^{2}$. Very recently, it was further shown in the work of Li et al. that there exist $q$ -ary optimal locally repairable codes with the length bigger than $q+1$ and the distance proportional to $n$. Thus, it becomes an interesting and challenging problem to construct new families of $q$ -ary optimal locally repairable codes of length bigger than $q+1$. In this paper, we construct a class of optimal locally repairable codes of distances 3 and 4 with unbounded length (i.e., length of the codes is independent of the code alphabet size). Our technique is through cyclic codes with particular generator and parity-check polynomials that are carefully chosen. [ABSTRACT FROM AUTHOR]

When transmitting information over a noisy channel, two approaches, dating back to Shannon’s work, are common: assuming the channel errors are independent of the transmitted content and devising an error-correcting code or assuming the errors are data dependent and devising a constrained-coding scheme that eliminates all offending data patterns. In this paper, we analyze a middle road, which we call a semiconstrained system. In such a system, which is an extension of the channel with the cost constraints model, we do not eliminate the error-causing sequences entirely, but rather restrict the frequency in which they appear. We address several key issues in this paper. The first is proving closed-form bounds on the capacity, which allow us to bound the asymptotics of the capacity. In particular, we bound the rate at which the capacity of the semiconstrained $(0,k)$ -RLL tends to 1 as $k$ grows. The second key issue is devising efficient encoding and decoding procedures that asymptotically achieve capacity with vanishing error. Finally, we consider delicate issues involving the continuity of the capacity and a relaxation of the definition of semiconstrained systems. [ABSTRACT FROM AUTHOR]

We use deterministic and probabilistic methods to analyze the performance of compressed sensing matrices constructed from Hadamard matrices and pairwise balanced designs, previously introduced by a subset of the authors. In this paper, we obtain upper and lower bounds on the sparsity of signals for which our matrices guarantee recovery. These bounds are tight to within a multiplicative factor of at most 4\sqrt 2 . We provide new theoretical results and detailed simulations, which indicate that the construction is competitive with Gaussian random matrices, and that recovery is tolerant to noise. A new recovery algorithm tailored to the construction is also given. [ABSTRACT FROM AUTHOR]

The noise model of deletions poses significant challenges in coding theory, with basic questions like the capacity of the binary deletion channel still being open. In this paper, we study the harder model of worst case deletions, with a focus on constructing efficiently decodable codes for the two extreme regimes of high-noise and high-rate. Specifically, we construct polynomial-time decodable codes with the following tradeoffs (for any \varepsilon > 0 ): 1) codes that can correct a fraction 1- \varepsilon of deletions with rate \mathop \mathrm poly\nolimits ( \varepsilon ) over an alphabet of size \mathop \mathrm poly\nolimits (1/ \varepsilon ) ; 2) binary codes of rate 1-\tilde O(\sqrt \varepsilon ) that can correct a fraction $ \varepsilon $ of deletions; and 3) Binary codes that can be list-decoded from a fraction (1/2- \varepsilon )$ of deletions with rate \mathop {\mathrm {poly}}\nolimits ( \varepsilon ) . This paper gives the first efficient constructions which meet the qualitative goals of correcting a deletion fraction approaching 1 over bounded alphabets, and correcting a constant fraction of bit deletions with rate approaching 1 over a fixed alphabet. The above-mentioned results bring our understanding of deletion code constructions in these regimes to a similar level as worst case errors. [ABSTRACT FROM PUBLISHER]

This paper studies problems in data reconstruction, an important area with numerous applications. In particular, we examine the reconstruction of binary and nonbinary sequences from synchronization (insertion/deletion-correcting) codes. These sequences have been corrupted by a fixed number of symbol insertions (larger than the minimum edit distance of the code), yielding a number of distinct traces to be used for reconstruction. We wish to know the minimum number of traces needed for exact reconstruction. This is a general version of a problem tackled by Levenshtein for uncoded sequences. We introduce an exact formula for the maximum number of common supersequences shared by sequences at a certain edit distance, yielding an upper bound on the number of distinct traces necessary to guarantee exact reconstruction. Without specific knowledge of the code words, this upper bound is tight. We apply our results to the famous single deletion/insertion-correcting Varshamov–Tenengolts (VT) codes and show that a significant number of VT code word pairs achieve the worst case number of outputs needed for exact reconstruction. We also consider extensions to other channels, such as adversarial deletion and insertion/deletion channels and probabilistic channels. [ABSTRACT FROM PUBLISHER]

An error-erasure channel is a simple noise model that introduces both errors and erasures. While the two types of errors can be corrected simultaneously with error-correcting codes, it is also known that any linear code allows for first correcting errors and then erasures in two-step decoding. In particular, a carefully designed parity-check matrix not only allows for separating erasures from errors but also makes it possible to efficiently correct erasures. The separating redundancy of a linear code is the number of parity-check equations in a smallest parity-check matrix that has the required property for this error-erasure separation. In a sense, it is a parameter of a linear code that represents the minimum overhead for efficiently separating erasures from errors. While several bounds on separating redundancy are known, there still remains a wide gap between upper and lower bounds except for a few limited cases. In this paper, using probabilistic combinatorics and design theory, we improve both upper and lower bounds on separating redundancy. We also show a relation between parity-check matrices for error-erasure separation and special matrices, called X-codes, for data compaction circuits in VLSI testing. This leads to an exponentially improved bound on the size of an optimal X-code. [ABSTRACT FROM AUTHOR]

Multi-permutations and in particular permutations appear in various applications in an information theory. New applications, such as rank modulation for flash memories, have suggested the need to consider error-correcting codes for multi-permutations. In this paper, we study systematic error-correcting codes for multi-permutations in general and for permutations in particular. For a given number of information symbols k , and for any integer t , we present a construction of (k+r,k) systematic t -error-correcting codes, for permutations of length $k+r$ , where the number of redundancy symbols $r$ is relatively small. In particular, for a given $t$ and for sufficiently large $k$ , we obtain $r=t+1$ , while a lower bound on the number of redundancy symbols is shown to be $t$ . The same construction is also applied to obtain related systematic error-correcting codes for any types of multi-permutations. [ABSTRACT FROM AUTHOR]

Kulkarni and Kiyavash recently introduced a new method to establish upper bounds on the size of deletion-correcting codes. This method is based upon tools from hypergraph theory. The deletion channel is represented by a hypergraph whose edges are the deletion balls (or spheres), so that a deletion-correcting code becomes a matching in this hypergraph. Consequently, a bound on the size of such a code can be obtained from bounds on the matching number of a hypergraph. Classical results in hypergraph theory are then invoked to compute an upper bound on the matching number as a solution to a linear-programming problem: the problem of finding fractional transversals. The method by Kulkarni and Kiyavash can be applied not only for the deletion channel but also for other error channels. This paper studies this method in its most general setup. First, it is shown that if the error channel is regular and symmetric then the upper bound by this method coincides with the well-known sphere packing bound and thus is called here the generalized sphere packing bound. Even though this bound is explicitly given by a linear programming problem, finding its exact value may still be a challenging task. The art of finding the exact upper bound (or slightly weaker ones) is the assignment of weights to the hypergraph’s vertices in a way that they satisfy the constraints in the linear programming problem. In order to simplify the complexity of the linear programming, we present a technique based upon graph automorphisms that in many cases significantly reduces the number of variables and constraints in the problem. We then apply this method on specific examples of error channels. We start with the $Z$ channel and show how to exactly find the generalized sphere packing bound for this setup. Next studied is the nonbinary limited magnitude channel both for symmetric and asymmetric errors, where we focus on the single-error case. We follow up on the deletion channel, which was the original motivation of the work by Kulkarni and Kiyavash, and show how to improve upon their upper bounds for single-deletion-correcting codes. Since the deletion and grain-error channels have a similar structure for a single error, we also improve upon the existing upper bounds on single-grain error-correcting codes. Finally, we apply this method for projective spaces and find its generalized sphere packing bound for the single-error case. [ABSTRACT FROM AUTHOR]

Frameproof codes are used to fingerprint digital data. They can prevent copyrighted materials from unauthorized use. In this paper, we study upper and lower bounds for $w$ -frameproof codes of length N$ over an alphabet of size q$ . The upper bound is based on a combinatorial approach and the lower bound is based on a probabilistic construction. Both bounds can improve one of the previous results when q$ is small compared with w$ , say cq\leq w$ for some constant c\leq q$ . Furthermore, we pay special attention to binary frameproof codes. We show a binary w . [ABSTRACT FROM PUBLISHER]

The ability to store data in the DNA of a living organism has applications in a variety of areas including synthetic biology and watermarking of patented genetically modified organisms. Data stored in this medium are subject to errors arising from various mutations, such as point mutations, indels, and tandem duplication, which need to be corrected to maintain data integrity. In this paper, we provide error-correcting codes for errors caused by tandem duplications, which create a copy of a block of the sequence and insert it in a tandem manner, i.e., next to the original. In particular, we present two families of codes for correcting errors due to tandem duplications of a fixed length: the first family can correct any number of errors, while the second corrects a bounded number of errors. We also study codes for correcting tandem duplications of length up to a given constant $k$ , where we are primarily focused on the cases of $k=2,3$ . Finally, we provide a full classification of the sets of lengths allowed in tandem duplication that result in a unique root for all sequences. [ABSTRACT FROM AUTHOR]

In this paper, we investigate properties of generalized bent Boolean functions and 2^k-bent (i.e., negabent, octabent, hexadecabent, et al.) Boolean functions in a uniform framework. From the Hadamard matrices, Hodžić and Pasalic presented sufficient conditions for generalized bent functions. Using cyclotomic fields and the decomposition of generalized bent functions, we generalize their results, prove that Hodžić and Pasalic’s conditions of generalized bent functions are not only sufficient but also necessary, and completely characterize generalized bent functions in terms of their component functions. Furthermore, we present a secondary construction of bent functions or semi-bent functions from generalized bent functions. Finally, we give the relations of generalized bent functions and 2^k-bent functions, demonstrate that 2^k-bent functions are actually a special class of generalized bent functions, and completely characterize 2^k-bent functions. [ABSTRACT FROM PUBLISHER]

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]

Symbol-pair read channels, in which the outputs of the read process are pairs of consecutive symbols, were recently studied by Cassuto and Blaum. This new paradigm is motivated by the limitations of the reading process in high density data storage systems. They studied error correction in this new paradigm, specifically, the relationship between the minimum Hamming distance of an error correcting code and the minimum pair distance, which is the minimum Hamming distance between symbol-pair vectors derived from codewords of the code. It was proved that for a linear cyclic code with minimum Hamming distance dH , the corresponding minimum pair distance is at least dH+3 . In this paper, we show that, for a given linear cyclic code with a minimum Hamming distance dH , the minimum pair distance is at least dH +\left \lceil{ {dH/{2}}}\right \rceil . We then describe a decoding algorithm, based upon a bounded distance decoder for the cyclic code, whose symbol-pair error correcting capabilities reflect the larger minimum pair distance. Finally, we consider the case where the read channel output is a larger number, $b \geqslant 3$ , of consecutive symbols, and we provide extensions of several concepts, results, and code constructions to this setting. [ABSTRACT FROM AUTHOR]

Motivated by the rank modulation scheme, a recent study by Sala and Dolecek explored the idea of constraint codes for permutations. The constraint studied by them is inherited by the inter-cell interference phenomenon in flash memories, where high-level cells can inadvertently increase the level of low-level cells. A permutation \sigma \in Sn satisfies the single-neighbor $k$ -constraint if \sigma (i+1)-\sigma (i)\leq k$ for all 1\leq i\leq n-1$ . In this paper, this model is extended into two constraints. A permutation k$ -constraint if for all 2 \leq i \leq n-1$ , \sigma (i)-\sigma (i-1)\leq k$ or \sigma (i \,\, + \,\, 1)-\sigma (i)\leq k$ , and it satisfies the asymmetric two-neighbor k$ -constraint if for all 2 \leq i \leq n-1$ , and the capacity of the second constraint is 1 regardless for any positive $k$ . We also extend our results and study the capacity of these two constraints combined with error-correcting codes in the Kendall $\tau $ -metric. [ABSTRACT FROM AUTHOR]

We study the problem of constructing balanced generator matrices for Reed–Solomon and Tamo–Barg codes. More specifically, we are interested in realizing generator matrices, for the full-length cyclic versions of these codes, where all rows have the same weight and the difference in weight between any columns is at most one. The results presented in this paper translate to computationally balanced encoding schemes, which can be appealing in distributed storage applications. Indeed, the balancedness of these generator matrices guarantees that the computation effort exerted by any storage node is essentially the same. In general, the framework presented can accommodate various values for the required row weight. We emphasize the possibility of constructing sparsest and balanced generator matrices for Reed–Solomon codes, i.e., each row is a minimum distance codeword. The number of storage nodes contacted once a message symbol is updated decreases with the row weight, so sparse constructions are appealing in that context. Results of similar flavor are presented for cyclic Tamo–Barg codes. In particular, we show that for a code with minimum distance $d$ and locality $r$ , a construction in which every row is of weight $d + r - 1$ is possible. The constructions presented are deterministic and operate over the codes’ original underlying finite field. As a result, efficient decoding from both errors and erasures is possible thanks to the plethora of efficient decoders available for the codes considered. [ABSTRACT FROM AUTHOR]

Quadratic residue codes are a well-known class of codes. In this paper, we consider the constructions of self-dual codes by higher power residues, especially fourth power residues. New infinite families of self-dual codes over GF(2), GF(3), GF(4), GF(8), and GF(9) are introduced. Some of them have better minimum weight than previously known codes. We also give general results related to the automorphism group of some of these codes. [ABSTRACT FROM PUBLISHER]

This paper provides new constructions and lower bounds for subspace codes, using Ferrers diagram rank-metric codes from matchings of the complete graph and pending blocks. We present different constructions for constant dimension codes with minimum injection distance 2 or $k-1$ , where $k$ is the constant dimension. Furthermore, we present a construction of new codes from old codes for any minimum distance. Then, we construct nonconstant dimension codes from these codes. Some examples of codes obtained by these constructions are the largest known codes for the given parameters. [ABSTRACT FROM PUBLISHER]

An efficient isomorph-free generation algorithm for classification of binary self-dual codes with the minimum distance $d = 4$ is presented. It is combined with another isomorph-free generation algorithm for classification of self-dual codes with the minimum distance $d \ge 6$ . A complete classification of all binary self-dual codes of length 40 is given as a result. [ABSTRACT FROM PUBLISHER]

Locally recoverable (LRC) codes have recently been a focus point of research in coding theory due to their theoretical appeal and applications in distributed storage systems. In an LRC code, any erased symbol of a codeword can be recovered by accessing only a small number of other symbols. For LRC codes over a small alphabet (such as binary), the optimal rate-distance trade-off is unknown. We present several new combinatorial bounds on LRC codes including the locality-aware sphere packing and Plotkin bounds. We also develop an approach to linear programming (LP) bounds on LRC codes. The resulting LP bound gives better estimates in examples than the other upper bounds known in the literature. Further, we provide the tightest known upper bound on the rate of linear LRC codes with a given relative distance, an improvement over the previous best known bounds. [ABSTRACT FROM PUBLISHER]

Many of the classic problems of the coding theory are highly symmetric, which makes it easy to derive the sphere-packing upper bounds on the size of codes. We discuss the generalizations of the sphere-packing bounds to the arbitrary error models. These generalizations become especially important when the sizes of the error spheres are nonuniform. The best possible sphere-packing bounds are the solutions to the linear programs. We derive a series of bounds from the approximations to the packing and covering problems and study the relationships and the trade-offs between them. We show the method to obtain the upper bounds by optimizing across a family of channels that admit the same codes. We present a generalization of the local degree bound of Kulkarni and Kiyavash, and use it to improve the best-known upper bounds on the sizes of the single deletion correcting codes and the single grain error correcting codes. [ABSTRACT FROM AUTHOR]

A covering code is a set of codewords with the property that the union of balls, suitably defined, around these codewords covers an entire space. Generally, the goal is to find the covering code with the minimum size codebook. While most prior work on covering codes has focused on the Hamming metric, we consider the problem of designing covering codes defined in terms of either insertions or deletions. First, we provide new sphere-covering lower bounds on the minimum possible size of such codes. Then, we provide new existential upper bounds on the size of optimal covering codes for a single insertion or a single deletion that are tight up to a constant factor. Finally, we derive improved upper bounds for covering codes using R ≥ 2 insertions or deletions. We prove that codes exist with density that is only a factor O(R log R) larger than the lower bounds for all fixed R. In particular, our upper bounds have an optimal dependence on the word length, and we achieve asymptotic density matching the best known bounds for Hamming distance covering codes. [ABSTRACT FROM AUTHOR]

We study covering codes of permutations with the $\ell _\infty $ -metric. We provide a general code construction, which combines short building-block codes into a single long code. We focus on cyclic transitive groups as building blocks, determining their exact covering radius, and showing a linear-time algorithm for finding a covering codeword. When used in the general construction, we show that the resulting covering code asymptotically out-performs the best known code while maintaining linear-time decoding. We also bound the covering radius of relabeled cyclic transitive groups under conjugation, showing that the covering radius is quite robust. While relabeling cannot reduce the covering radius by much, the downside is that we prove the covering radius cannot be increased by more than 1 when using relabeling. [ABSTRACT FROM AUTHOR]

We investigate deletion correcting codes and restricted composition codes in particular. Restricted composition codes generalize codes on permutations and multipermutations. We use graph theoretic methods to characterize codes, establish bounds on code size, and describe constructions. The substring partial order has a well-known property. For any string, the number of superstrings of a particular length depends only on the length of the original string. We generalize this property to take compositions into account. For any string, the number of superstrings of a particular composition depends only on the composition of the original string. We present a bijective proof of this fact. We apply this result to obtain a lower bound on the size of restricted composition codes. We obtain an upper bound by analyzing deletion errors when the composition of deleted symbols is restricted to a particular worst case composition. We construct binary restricted composition single-deletion correcting codes and show that they are asymptotically optimal and form an optimal coloring. There is a natural distance on compositions that provides a lower bound on deletion distance. Unrestricted deletion correcting codes can be constructed from the union of restricted composition codes as long as the set of compositions used themselves form a code. The nonbinary single-deletion correcting codes constructed by Tenengolts are a special case of our method. [ABSTRACT FROM PUBLISHER]

Quantum maximum-distance-separable (MDS) codes form an important class of quantum codes. To get q -ary quantum MDS codes, one of the effective ways is to find linear MDS codes C over \mathbb Fq^{2} satisfying C^{\perp H}\subseteq C , where C^\perp H denotes the Hermitian dual code of C . For a linear code C of length n over , we say that C is a dual-containing code if C^\perp H\subseteq C and C\neq \mathbb {F}q^{2}^{n} . Several classes of new quantum MDS codes with relatively large minimum distance have been produced through dual-containing constacyclic MDS codes. These works motivate us to make a careful study on the existence conditions for dual-containing constacyclic codes. We obtain necessary and sufficient conditions for the existence of dual-containing constacyclic codes. Four classes of dual-containing constacyclic MDS codes are constructed and their parameters are computed. Consequently, the quantum MDS codes are derived from these parameters. The quantum MDS codes exhibited here have minimum distance bigger than the ones available in the literature. [ABSTRACT FROM PUBLISHER]