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We study codes with parameters of $q$ -ary shortened Hamming codes, i.e., $(n=(q^{m}-q)/(q-1), q^{n-m}, 3)_{q}$. Firstly, we prove the fact mentioned in 1998 by Brouwer et al. that such codes are optimal, generalizing it to a bound for multifold packings of radius-1 balls, with a corollary for multiple coverings. In particular, we show that the punctured Hamming code is an optimal $q$ -fold packing with minimum distance 2. Secondly, for every admissible length starting from $n=20$ , we show the existence of 4-ary codes with parameters of shortened 1-perfect codes that cannot be obtained by shortening a 1-perfect code. [ABSTRACT FROM AUTHOR]

Let $q\geq 2$ be a prime power. In this paper, we study the distribution of the minimum distance (in the Hamming metric) of a random linear code of dimension $k$ in $\mathbb {F}_{q}^{n}$. We provide quantitative estimates showing that the distribution function of the minimum distance is close (superpolynomially in $n$) to the cumulative distribution function of the minimum of $(q^{k}-1)/(q-1)$ independent binomial random variables with parameters $\frac {1}{q}$ and $n$. The latter, in turn, converges to a Gumbel distribution at integer points when $\frac {k}{n}$ converges to a fixed number in (0, 1). Our result confirms in a strong sense that apart from identification of the weights of proportional codewords, the probabilistic dependencies introduced by the linear structure of the random code, produce a negligible effect on the minimum code weight. As a corollary of the main result, we obtain an improvement of the Gilbert–Varshamov bound for $2< q< 49$. [ABSTRACT FROM AUTHOR]

Trace reconstruction considers the task of recovering an unknown string $\mathbf {x}\in \{0,1\}^{n}$ given a number of independent “traces”, i.e., subsequences of $\mathbf {x}$ obtained by randomly and independently deleting every symbol of $\mathbf {x}$ with some probability $p$. The information-theoretic limit of the number of traces needed to recover a string of length $n$ is still unknown. This limit is essentially the same as the number of traces needed to determine, given strings $\mathbf {x}$ and $\mathbf {y}$ and traces of one of them, which string is the source. The most-studied class of algorithms for the worst-case version of the problem are “mean-based” algorithms. These are a restricted class of distinguishers that only use the mean value of each coordinate on the given samples. In this work we study limitations of mean-based algorithms on strings at small Hamming or edit distance. We show that, on the one hand, distinguishing strings that are nearby in Hamming distance is “easy” for such distinguishers. On the other hand, we show that distinguishing strings that are nearby in edit distance is “hard” for mean-based algorithms. Along the way, we also describe a connection to the famous Prouhet-Tarry-Escott (PTE) problem, which shows a barrier to finding explicit hard-to-distinguish strings: namely such strings would imply explicit short solutions to the PTE problem, a well-known difficult problem in number theory. Furthermore, we show that the converse is also true, thus, finding explicit solutions to the PTE problem is equivalent to the problem of finding explicit strings that are hard-to-distinguish by mean-based algorithms. Our techniques rely on complex analysis arguments that involve careful trigonometric estimates, and algebraic techniques that include applications of Descartes’ rule of signs for polynomials over the reals. [ABSTRACT FROM AUTHOR]

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

We study the Singleton-type bound that provides an upper limit on the minimum distance of locally repairable codes. We present an improved bound by carefully analyzing the combinatorial structure of the repair sets. Thus, we show the previous bound is unachievable for certain parameters. We then also provide explicit constructions of optimal codes which show that for certain parameters the new bound is sharp. Additionally, as a byproduct, some previously known codes are shown to attain the new bound and are thus proved to be optimal. [ABSTRACT FROM AUTHOR]

The six infinite families of power APN functions are among the oldest known instances of APN functions, and it has been conjectured in 2000 that they exhaust all possible power APN functions. Another long-standing open problem is that of the Walsh spectrum of the Dobbertin power family, which is still unknown. Those of Kasami, Niho and Welch functions are known, but not the precise values of their Walsh transform, with rare exceptions. One promising approach that could lead to the resolution of these problems is to consider alternative representations of the functions in questions. We derive alternative representations for the infinite APN monomial families. We show how the Niho, Welch, and Dobbertin functions can be represented as the composition $x^{i} \circ x^{1/j}$ of two power functions, and prove that our representations are optimal, i.e. no two power functions of lesser algebraic degree can be used to represent the functions in this way. We investigate compositions $x^{i} \circ L \circ x^{1/j}$ for a linear polynomial $L$ , show how the Kasami functions in odd dimension can be expressed in this way with $i=j$ being a Gold exponent and compute all APN functions of this form for $n \le 9$ and for $L$ with binary coefficients, thereby showing that our theoretical constructions exhaust all possible cases. We present observations and data on power functions with exponent $\sum _{i = 1}^{k-1} 2^{2ni} - 1$ which generalize the inverse and Dobbertin families. We present data on the Walsh spectrum of the Dobbertin function for $n \le 35$ , and conjecture its exact form. As an application of our results, we determine the exact values of the Walsh transform of the Kasami function at all points of a special form. Computations performed for $n\leq 21$ show that these points cover about 2/3 of the field. [ABSTRACT FROM AUTHOR]

Insertion–deletion codes (insdel codes for short) play an important role in synchronization error correction. The higher the minimum insdel distance, the more insdel errors the code can correct. Haeupler and Shahrasbi established the Singleton bound for insdel codes: the minimum insdel distance of any $[n,k]$ linear code over $\mathbb {F}_{q}$ satisfies $d\leq 2n-2k+2$. There have been some constructions of insdel codes through Reed-Solomon codes with high capabilities, but none has come close to this bound. Recently, Do Duc et al. showed that the minimum insdel distance of any $[n,k]$ Reed-Solomon code is no more than $2n-2k$ if $q$ is large enough compared to the code length $n$ ; optimal codes that meet the new bound were also constructed explicitly. The contribution of this paper is twofold. We first show that the minimum insdel distance of any $[n,k]$ linear code over $\mathbb {F}_{q}$ satisfies $d\leq 2n-2k$ if $n > k > 1$. This result improves and generalizes the previously known results in the literature. We then give a sufficient condition under which the minimum insdel distance of a 2-D Reed-Solomon code of length $n$ over $\mathbb {F}_{q}$ is exactly equal to $2n-4$. As a consequence, we show that the sufficient condition is not hard to achieve; we explicitly construct an infinite family of optimal 2-D Reed-Somolom codes meeting the bound. [ABSTRACT FROM AUTHOR]

We study $k$ -uniform states in heterogeneous systems whose local dimensions are not all the same. Based on the connections between mixed orthogonal arrays with certain minimum Hamming distance, irredundant mixed orthogonal arrays and $k$ -uniform states, we present two constructions of 2-uniform states in heterogeneous systems. We also construct two families of 3-uniform states in heterogeneous systems, which solves a question posed in [D. Goyeneche et al., Phys. Rev. A 94, 012346 (2016)]. We show two methods of generating $(k-1)$ -uniform states from $k$ -uniform states. Some results on the nonexistence of absolutely maximally entangled states are provided. For the applications, we present an orthogonal basis consisting of $k$ -uniform states with the minimum support, and we show that some $k$ -uniform bases are locally irreducible. Moreover, we connect $k$ -uniform states with quantum information masking. [ABSTRACT FROM AUTHOR]

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

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

In this paper, we introduce the notion of generalized pair weights of an $[n, k]$ -linear code over the finite field $\mathbb {F}_{q}$ and the notion of pair $r$ -equiweight codes, where $1\le r\le k-1$. Some basic properties of generalized pair weights of linear codes over finite fields are derived. We obtain a necessary and sufficient condition for an $[n,k]$ -linear code to be a pair equiweight code, and we characterize pair $r$ -equiweight codes for any $1\le r\le k-1$. A necessary and sufficient condition for a linear isomorphism to preserve pair weights between two linear codes is obtained. At the end of this paper, an application of generalized pair weights of linear codes to symbol-pair read wire-tap channels of type II is introduced. [ABSTRACT FROM AUTHOR]

We construct maximally recoverable codes (corresponding to partial MDS codes) which are based on linearized Reed-Solomon codes. The new codes have a smaller field size requirement compared with known constructions. For certain asymptotic regimes, the constructed codes have order-optimal alphabet size, asymptotically matching the known lower bound. [ABSTRACT FROM AUTHOR]

We revisit and take a closer look at a (not so well known) result of a 2017 paper, showing that the differential uniformity of any vectorial function is bounded from below by an expression depending on the size of its image set. We make explicit the resulting tight lower bound on the image set size of differentially $\delta $ -uniform functions (which is the only currently known non-trivial lower bound on the image set size of such functions). We also significantly improve an upper bound on the nonlinearity of vectorial functions obtained in the same reference and involving their image set size. We study when the resulting bound is sharper than the covering radius bound. We obtain as a by-product a lower bound on the Hamming distance between differentially $\delta $ -uniform functions and affine functions, which we improve significantly with a second bound. This leads us to study what can be the maximum Hamming distance between vectorial functions and affine functions. We provide an upper bound which is slightly sharper than a bound by Liu, Mesnager and Chen when $m < n$ , and a second upper bound, which is much stronger in the case (happening in practice) where $m$ is near $n$ ; we study the tightness of this latter bound; this leads to an interesting question on APN functions, which we address (negatively). We finally derive an upper bound on the nonlinearity of vectorial functions by means of their Hamming distance to affine functions and make more precise the bound on the differential uniformity which was the starting point of the paper. [ABSTRACT FROM AUTHOR]

A private proximity retrieval (PPR) scheme is a protocol which allows a user to retrieve the identities of all records in a database that are within some distance $r$ from the user’s record $x$. The user’s privacy at each server is given by the fraction of the record $x$ that is kept private. In this paper, this research is initiated and protocols that offer trade-offs between privacy, computational complexity, and storage are studied. In particular, we assume that each server stores a copy of the database and study the required minimum number of servers by our protocol which provides a given privacy level. Each server receives a query in the protocol and the set of queries forms a code. The main focus in the paper is dedicated to studying the family of codes generated by the set of queries. These codes will be shown to satisfy a specific covering property and will be called private proximity retrieval intersection covering codes. In particular, since the query every server receives is a codeword, the goal is to minimize the number of codewords in such a code which is the minimum number of servers required by the protocol. These codes are closely related to a family of codes known as covering designs. We introduce several lower bounds on the sizes of such codes as well as several constructions. This work focuses on the case when the records are binary vectors together with the Hamming distance. Other metrics such as the Johnson metric are also investigated. [ABSTRACT FROM AUTHOR]

A subset of a vector space $\mathbb {F}_{q}^{n}$ is additive if it is a linear space over the field $\mathbb {F}_{p}$ , where $q=p^{e}$ , $p$ prime, and $e>1$. Bounds on the rank and dimension of the kernel of additive generalised Hadamard (additive GH) codes are established. For specific ranks and dimensions of the kernel within these bounds, additive GH codes are constructed. Moreover, for the case $e=2$ , it is shown that the given bounds are tight and it is possible to construct an additive GH code for all allowable ranks and dimensions of the kernel between these bounds. Finally, we also prove that these codes are self-orthogonal with respect to the trace Hermitian inner product, and generate pure quantum codes. [ABSTRACT FROM AUTHOR]

We initiate a study, when $F$ is a general APN function, of the Boolean function $\gamma _{F}$ related to the differential spectrum of $F$ (and which is known to be bent if and only if $F$ is almost bent). We first list many open questions about it. We study its algebraic normal form and its bivariate representation. We characterize its linear structures and specify nonexistence cases; we show, for $n$ even, their relation with the bent components of $F$. We pose three related open problems. We characterize further in terms of $\gamma _{F}$ the fact that a component function of $F$ is bent and study if the number of bent components can be optimal. We consider in particular two classes, one of which is that of APN power functions. We study more deeply the relation between the Walsh transform of $\gamma _{F}$ and the Walsh transform of $F$. By applying the Titsworth relation to the Walsh transform $W_{\gamma _{F}}$ , we deduce a new relation satisfied by $W_{F}^{2}$ , which is as simple as Chabaud-Vaudenay’s characterization by the fourth moment of the Walsh transform (which is in fact a particular case of the new relation), and provides more information. From this new relation, we deduce, for a sub-class of APN functions, a lower bound on the nonlinearity, which is significantly stronger than $nl(F)>0$ (the only general known bound). This sub-class of APN functions includes all known APN functions. The question (which is another open problem that we state) arises whether this sub-class equals that of all APN functions, but our bound provides at least a beginning of explanation why all known APN functions have non-weak nonlinearity. We finally show how the nonlinearities of $\gamma _{F}$ and $F$ are related by a simple formula; this leads to a last open problem. [ABSTRACT FROM AUTHOR]

We propose a generalization of the Wasserstein distance of order 1 to the quantum states of $n$ qudits. The proposal recovers the Hamming distance for the vectors of the canonical basis, and more generally the classical Wasserstein distance for quantum states diagonal in the canonical basis. The proposed distance is invariant with respect to permutations of the qudits and unitary operations acting on one qudit and is additive with respect to the tensor product. Our main result is a continuity bound for the von Neumann entropy with respect to the proposed distance, which significantly strengthens the best continuity bound with respect to the trace distance. We also propose a generalization of the Lipschitz constant to quantum observables. The notion of quantum Lipschitz constant allows us to compute the proposed distance with a semidefinite program. We prove a quantum version of Marton’s transportation inequality and a quantum Gaussian concentration inequality for the spectrum of quantum Lipschitz observables. Moreover, we derive bounds on the contraction coefficients of shallow quantum circuits and of the tensor product of one-qudit quantum channels with respect to the proposed distance. We discuss other possible applications in quantum machine learning, quantum Shannon theory, and quantum many-body systems. [ABSTRACT FROM AUTHOR]

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]

This paper studies reconstruction of strings based upon their substrings spectrum. Under this paradigm, it is assumed that all substrings of some fixed length are received and the goal is to reconstruct the string. While many existing works assumed that substrings are received error free, we follow in this paper the noisy setup of this problem that was first studied by Gabrys and Milenkovic. The goal of this study is twofold. First we study the setup in which not all substrings in the multispectrum are received, and then we focus on the case where the read substrings are not error free. In each case we provide specific code constructions of strings that their reconstruction is guaranteed even in the presence of failure in either model. We present efficient encoding and decoding maps and analyze the cardinality of the code constructions, while studying the cases where the rates of our codes approach 1. [ABSTRACT FROM AUTHOR]

This paper revisits the ordered statistics decoding (OSD). It provides a comprehensive analysis of the OSD algorithm by characterizing the statistical properties, evolution and the distribution of the Hamming distance and weighted Hamming distance from codeword estimates to the received sequence in the reprocessing stages of the OSD algorithm. We prove that the Hamming distance and weighted Hamming distance distributions can be characterized as mixture models capturing the decoding error probability and code weight enumerator. Simulation and numerical results show that our proposed statistical approaches can accurately describe the distance distributions. Based on these distributions and with the aim to reduce the decoding complexity, several techniques, including stopping rules and discarding rules, are proposed, and their decoding error performance and complexity are accordingly analyzed. Simulation results for decoding various eBCH codes demonstrate that the proposed techniques can significantly reduce the decoding complexity with a negligible loss in the decoding error performance. [ABSTRACT FROM AUTHOR]

Motivated by the duplication-correcting problem for data storage in live DNA, we study the construction of constant-weight codes in ℓ1-metric. By using packings and group divisible designs in combinatorial design theory, we give constructions of optimal codes over non-negative integers and optimal ternary codes with ℓ1-weight w ≤ 4 for all possible distances. In general, we derive the size of the largest ternary code with constant weight w and distance 2w−2 for sufficiently large length n satisfying n ≡ 1, w, −w+2,−2w+3 (mod w(w−1)). [ABSTRACT FROM AUTHOR]

This work is concerned with the problem of recovering high-dimensional signals x ∈ Rn which belong to a convex set of low complexity from a small number of quantized measurements. We propose to estimate the signals via a convex program based on rectified linear units (ReLUs) for two different quantization schemes, namely one-bit and uniform multi-bit quantization. Assuming that the linear measurement process can be modelled by a sensing matrix with i.i.d. subgaussian rows, we obtain for both schemes near-optimal uniform reconstruction guarantees by adding well-designed noise to the linear measurements prior to the quantization step. In the one-bit case, we show that the program is robust against adversarial bit corruptions as well as additive noise on the linear measurements. Further, our analysis quantifies precisely how the rate-distortion relationship of the program changes depending on whether we seek reconstruction accuracies above or below the level of additive noise. The proofs rely on recent results by Dirksen and Mendelson on non-Gaussian hyperplane tessellations. Finally, we complement our theoretical analysis with numerical experiments which compare our method to other state-of-the-art methodologies. [ABSTRACT FROM AUTHOR]

Symbol-pair codes were introduced by Cassuto and Blaum in 2010 to protect pair errors in symbol-pair read channels. Recently Yaakobi, Bruck and Siegel (2016) generalized this notion to b-symbol codes in order to consider consecutive b errors for a prescribed integer b ≥ 2, and they gave constructions and decoding algorithms. Cyclic codes were considered by various authors as candidates for symbol-pair codes and they established minimum distance bounds on (certain) cyclic codes. In this paper we use algebraic curves over finite fields in order to obtain tight lower and upper bounds on b-symbol Hamming weights of arbitrary cyclic codes over Fq. Here b ≥ 2 is an arbitrary prescribed positive integer and Fq is an arbitrary finite field. We also present a stability theorem for an arbitrary cyclic code C of dimension k and length n: the b-symbol Hamming weight enumerator of C is the same as the k-symbol Hamming weight enumerator of C if k ≤ b ≤ n−1. Moreover, we give improved tight lower and upper bounds on b-symbol Hamming weights of some cyclic codes related to irreducible cyclic codes. Throughout the paper the length n is coprime to q. [ABSTRACT FROM AUTHOR]

We obtain a characterization on self-orthogonality for a given binary linear code in terms of the number of column vectors in its generator matrix, which extends the result of Bouyukliev et al. (2006). As an application, we give an algorithmic method to embed a given binary k-dimensional linear code C (k = 3,4) into a self-orthogonal code of the shortest length which has the same dimension k and minimum distance d' ≥ d(C). For k > 4, we suggest a recursive method to embed a k -dimensional linear code to a self-orthogonal code. We also give new explicit formulas for the minimum distances of optimal self-orthogonal codes for any length n with dimension 4 and any length n ≢ 6, 13,14,21,22,28,29 (mod 31) with dimension 5. [ABSTRACT FROM AUTHOR]

Because of its high data density and longevity, DNA is emerging as a promising candidate for satisfying increasing data storage needs. Compared to conventional storage media, however, data stored in DNA is subject to a wider range of errors resulting from various processes involved in the data storage pipeline. In this article, we consider correcting duplication errors for both exact and noisy tandem duplications of a given length k. An exact duplication inserts a copy of a substring of length k of the sequence immediately after that substring, e.g., ACGT → ACG ACGT, where k = 3, while a noisy duplication inserts a copy suffering from substitution noise, e.g., ACGT → ACG A RedTGT. Specifically, we design codes that can correct any number of exact duplication and one noisy duplication errors, where in the noisy duplication case the copy is at Hamming distance 1 from the original. Our constructions rely upon recovering the duplication root of the stored codeword. We characterize the ways in which duplication errors manifest in the root of affected sequences and design efficient codes for correcting these error patterns. We show that the proposed construction is asymptotically optimal, in the sense that it has the same asymptotic rate as optimal codes correcting exact duplications only. [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]

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]

Optimal locally repairable codes with information locality are considered. Optimal codes are constructed, whose length is also order-optimal with respect to a new bound on the code length derived in this article. The length of the constructed codes is super-linear in the alphabet size, which improves upon the well known pyramid codes, whose length is only linear in the alphabet size. The recoverable erasure patterns are also analyzed for the new codes. Based on the recoverable erasure patterns, we construct generalized sector-disk (GSD) codes, which can recover from disk erasures mixed with sector erasures in a more general setting than known sector-disk (SD) codes. Additionally, the number of sectors in the constructed GSD codes is super-linear in the alphabet size, compared with known SD codes, whose number of sectors is only linear in the alphabet size. [ABSTRACT FROM AUTHOR]

We consider the algebraic combinatorics of the set of injections from a $k$ -element set to an $n$ -element set. In particular, we give a new combinatorial formula for the spherical functions of the Gelfand pair $(S_{k} \times S_{n}, diag(S_{k}) \times S_{n-k})$. We use this combinatorial formula to give new Delsarte linear programming bounds on the size of codes over injections. [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]

It was demonstrated that, as a nonlinear implementation of Slepian-Wolf Coding, Distributed Arithmetic Coding (DAC) outperforms traditional Low-Density Parity-Check (LPDC) codes for short code length and biased sources. This fact triggers research efforts into theoretical analysis of DAC. In our previous work, we proposed two analytical tools, Codebook Cardinality Spectrum (CCS) and Hamming Distance Spectrum, to analyze DAC for independent and identically-distributed (i.i.d.) binary sources with uniform distribution. This article extends our work on CCS from uniform i.i.d. binary sources to biased i.i.d. binary sources. We begin with the final CCS and then deduce each level of CCS backwards by recursion. The main finding of this article is that the final CCS of biased i.i.d. binary sources is not uniformly distributed over [0, 1). This article derives the final CCS of biased i.i.d. binary sources and proposes a numerical algorithm for calculating CCS effectively in practice. All theoretical analyses are well verified by experimental results. [ABSTRACT FROM AUTHOR]

The process of DNA-based data storage (DNA storage for short) can be mathematically modelled as a communication channel, termed DNA storage channel, whose inputs and outputs are sets of unordered sequences. To design error correcting codes for DNA storage channel, a new metric, termed the sequence-subset distance, is introduced, which generalizes the Hamming distance to a distance function defined between any two sets of unordered vectors and helps to establish a uniform framework to design error correcting codes for DNA storage channel. We further introduce a family of error correcting codes, referred to as sequence-subset codes, for DNA storage and show that the error-correcting ability of such codes is completely determined by their minimum distance. We derive some upper bounds on the size of the sequence-subset codes including a tight bound for a special case, a Singleton-like bound and a Plotkin-like bound. We also propose some constructions, including an optimal construction for that special case, which imply lower bounds on the size of such codes. [ABSTRACT FROM AUTHOR]

In this work, we study the problem of list decoding of insertions and deletions. We present a Johnson-type upper bound on the maximum list size. The bound is meaningful only when insertions occur. Our bound implies that there are binary codes of rate $\Omega (1)$ that are list-decodable from a 0.707-fraction of insertions. For any $\tau _{\mathsf {I}} \geq 0$ and $\tau _{\mathsf {D}} \in [0,1)$ , there exist $q$ -ary codes of rate $\Omega (1)$ that are list-decodable from a $\tau _{\mathsf {I}}$ -fraction of insertions and $\tau _{\mathsf {D}}$ -fraction of deletions, where $q$ depends only on $\tau _{\mathsf {I}}$ and $\tau _{\mathsf {D}}$. We also provide efficient encoding and decoding algorithms for list-decoding from $\tau _{\mathsf {I}}$ -fraction of insertions and $\tau _{\mathsf {D}}$ -fraction of deletions for any $\tau _{\mathsf {I}} \geq 0$ and $\tau _{\mathsf {D}} \in [0,1)$. Based on the Johnson-type bound, we derive a Plotkin-type upper bound on the code size in the Levenshtein metric. [ABSTRACT FROM AUTHOR]

The weight spectra of MDS codes of length n and dimension k over arbitrary alphabets are studied. For all q-ary MDS codes of dimension k and length ${n}\ne {q}+{k}-1 $ containing the zero codeword, it is shown that all k weights from n to ${n}-{k}+1 $ are realized. The weight spectrum in the remaining case ${n}={q}+{k}-1 $ is also determined. Additionally, we prove that all binary MDS codes are equivalent to linear MDS codes. The proofs are combinatorial, and self contained. [ABSTRACT FROM AUTHOR]

To equip DNA-based data storage with random-access capabilities, Yazdi et al. (2018) prepended DNA strands with specially chosen address sequences called primers and provided certain design criteria for these primers. We provide explicit constructions of error-correcting codes that are suitable as primer addresses and equip these constructions with efficient encoding algorithms. Specifically, our constructions take cyclic or linear codes as inputs and produce sets of primers with similar error-correcting capabilities. Using certain classes of BCH codes, we obtain infinite families of primer sets of length $n$ , minimum distance $d$ with $(d+1) \log _{4}\,\,n +O(1)$ redundant symbols. Our techniques involve reversible cyclic codes (1964), an encoding method of Tavares et al. (1971) and Knuth’s balancing technique (1986). In our investigation, we also construct efficient and explicit binary balanced error-correcting codes and codes for DNA computing. [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]

An additive quaternary [n,k,d]-code (length n, quaternary dimension k, minimum distance d) is a 2k-dimensional F2-vector space of n-tuples with entries in F2 ⊕ F2 (the 2-dimensional vector space over F2) with minimum Hamming distance d. We determine the optimal parameters of additive quaternary codes of dimension k ≤ 3. The most challenging case is dimension k = 2.5. We prove that an additive quaternary [n,2.5,d]-code where d < n−1 exists if and only if 3(n−d) ≥ ⌈d/2⌉ + ⌈d/4⌉ + ⌈d/8⌉. In particular, we construct new optimal 2.5-dimensional additive quaternary codes. As a by-product, we give a direct proof for the fact that a binary linear [3m,5,2e]2-code for e < m−1 exists if and only if the Griesmer bound 3(m−e) ≥e/2⌉ + ⌈e/4⌉ + ⌈e/8⌉ is satisfied. [ABSTRACT FROM AUTHOR]

In this paper we prove new lower bounds for the maximal size of permutation codes by connecting the theory of permutation codes with the theory of linear block codes. More specifically, using the columns of a parity check matrix of an $[{\it{ n,k,d}}]_{ {q}}$ linear block code, we are able to prove the existence of a permutation code in the symmetric group of degree n, having minimum distance at least d and large cardinality. With our technique, we obtain new lower bounds for permutation codes that enhance the ones in the literature and provide asymptotic improvements in certain regimes of length and distance of the permutation code. [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]

Recently, some infinite families of binary minimal and optimal linear codes were constructed from simplicial complexes by Hyun et al. Inspired by their work, we present two new constructions of codes over the ring $\mathbb {F}_{2}+u\mathbb {F}_{2}$ by employing simplicial complexes. When the simplicial complexes are all generated by a maximal element, we determine the Lee weight distributions of two classes of the codes over $\mathbb {F}_{2}+u\mathbb {F}_{2}$. Our results show that the codes have few Lee weights. Via the Gray map, we obtain an infinite family of binary codes meeting the Griesmer bound and a class of binary distance optimal codes. [ABSTRACT FROM AUTHOR]

In general, array codes consist of $m\times n$ arrays and in many cases, the arrays satisfy parity constraints along lines of different slopes (generally with a toroidal topology). Such codes are useful for RAID type of architectures, since they allow to replace finite field operations by XORs. We present expansions to traditional array codes of this type, like Blaum-Roth (BR) and extended EVENODD codes, by adding parity on columns. This vertical parity allows for recovery of one or more symbols in a column locally, i.e., by using the remaining symbols in the column without invoking the rest of the array. Properties and applications of the new codes are discussed, in particular to Locally Recoverable (LRC) codes. [ABSTRACT FROM AUTHOR]

Let $\Bbb F_{q}$ be a finite field with $q$ elements and $p$ be a prime with $\gcd (p,q)=1$. Let $G$ be a finite abelian $p$ -group and $\Bbb F_{q}(G)$ be a group algebra. In this paper, we find all primitive idempotents and minimal abelian group codes in the group algebra $\Bbb F_{q}(G)$. Furthermore, we give all LCD abelian codes (linear code with complementary dual) and self-orthogonal abelian codes of $\Bbb F_{q}(G)$. [ABSTRACT FROM AUTHOR]

Cassuto and Blaum recently studied the symbol-pair channel, a model where every two consecutive symbols are read together. This special channel structure is motivated by the limitations of the reading process in high density data storage systems, where it is no longer possible to read individual symbols. In this new paradigm, the errors are not individual symbol errors, but rather symbol-pair errors, where at least one of the symbols is erroneous. In this work, we study bounds and constructions of codes over the symbol-pair channel. We extend the Johnson bound and the linear programming bound for this channel and show that they improve upon existing bounds. We then propose new code constructions that improve upon existing results for pair-distance six, seven, and ten. [ABSTRACT FROM AUTHOR]

Given a ${q}$ -ary frequency hopping sequence set of length ${n}$ and size ${M}$ with Hamming correlation ${H}$ , one can obtain a ${q}$ -ary (nonlinear) cyclic code of length ${n}$ and size nM with Hamming distance n-H. Thus, every upper bound on the size of a code from coding theory gives an upper bound on the size of a frequency hopping sequence set. Indeed, all upper bounds from coding theory have been converted to upper bounds on frequency hopping sequence sets. On the other hand, a lower bound from coding theory does not automatically produce a lower bound for frequency hopping sequence sets. In particular, the most important lower bound, the Gilbert-Varshamov bound in coding theory, has not been transformed to a valid lower bound on frequency hopping sequence sets. The purpose of this paper is to transform the Gilbert-Varshamov bound from coding theory to frequency hopping sequence sets by establishing a connection between a special family of cyclic codes (which are called hopping cyclic codes in this paper) and frequency hopping sequence sets. We provide two proofs of the Gilbert-Varshamov bound. One is based on a probabilistic method that requires advanced tool–martingale. This proof covers the whole rate region. Another proof is purely elementary but only covers part of the rate region. [ABSTRACT FROM AUTHOR]

We prove a new upper bound on the size of codes $C \subseteq \{1,2,3,4\}^{n}$ with the property that every four distinct codewords in $C$ have a coordinate where they all differ. Specifically, we provide a self-contained proof that such codes have size at most $2^{6n/19 + o(n)}$ , that is, rate bounded asymptotically by 6/19 ≤ 0.3158 (measured in bits). This improves the previous best upper bound of 0.3512 due to (Arikan 1994), which in turn improved the 0.375 bound that followed from general bounds for perfect hashing due to (Fredman and Komlós, 1984) and (Körner and Marton, 1988). Finally, using a combination of our approach with a simple idea which exploits powerful bounds on the minimum distance of codes in the Hamming space, we further improve the upper bound to 0.31477. [ABSTRACT FROM AUTHOR]

Let p be an odd prime, s and m be positive integers. Cyclic codes of length $2p^{{s}}$ over $\mathbb {F}_{{p}^{{m}}}$ are the ideals $\langle ({x}-1)^{i}({x}+1)^{j} \rangle $ , where $0 \le {i}, {j} \le {p}^{ {s}}$ , of the principal ideal ring $\mathbb {F}_{{p}^{{m}}}[{x}]/\langle {x}^{2\textit {p}^{\textit {s}}}-1 \rangle $. Using this structure, the symbol-pair distances of all cyclic codes of length $2\textit {p}^{\textit {s}}$ over $\mathbb F_{{p}^{{m}}}$ are completely determined. In addition, we establish all MDS symbol-pair cyclic codes of length $2\textit {p}^{\textit {s}}$ over $\mathbb F_{{p}^{{m}}}$. Some MDS symbol-pair cyclic codes are better than all the known ones. Among others, we discuss possible applications to construct quantum MDS symbol-pair codes. [ABSTRACT FROM AUTHOR]

Locally repairable codes (LRC) have been studied from two approaches to locally repair multiple failed nodes: 1) parallel approach, in which a coordinate $i$ of an $[n,k,d]$ linear code is said to have locality $r$ and availability $t$ if there exist $t$ disjoint repair sets each of which contains at most $r$ other coordinates that can recover the value of the $i$ -th coordinate; 2) sequential approach, in which the erased symbols (failed nodes) are repaired, one by one, and any previously repaired node can be used to repair the remaining failed nodes. In this paper, we first consider LRC aiming at joint sequential-parallel repairing multiple failed nodes, and study the $(n,k,r,t,u)$ -ELRCs (Exact locally repairable codes) which are $[n,k]$ linear codes with the property that any set of failed nodes of size at most $t$ can be simultaneously repaired in parallel mode, and each element of a set $E$ of failed nodes of size at most $u$ can be sequentially repaired by $r$ ($r< k$) other coordinates. We present a method by which with a given parity-check matrix of an $(n,k,r,t,u)$ -ELRC with minimum Hamming distance $d$ , a new ELRC with minimum Hamming distance $2d$ and availability $t+1$ is constructed that can repair each set of failed nodes $E$ of size at most $2u+1$ in sequential mode and this repair is done in at most $u-t+2$ steps. We construct a big family of LRCs by making use of orthogonal Latin rectangles and permutation cubes and some other combinatorial designs; the constructed codes contain the family of direct product codes; we also use $m$ -dimensional permutation cubes to construct LRCs with short block length for each $r$. [ABSTRACT FROM AUTHOR]

The error linear complexity spectrum constitutes a well-known cryptographic criterion for sequences, indicating how the linear complexity of the sequence decreases as the number of bits allowed to be modified per period increases. In this paper, via defining an association between $2^{n}$ -periodic binary sequences and Boolean functions on $n$ variables, it is shown that the error linear complexity spectrum also provides useful cryptographic information for the corresponding Boolean function $f$ - namely, it yields an upper bound on the minimum Hamming distance between $f$ and the set of functions depending on fewer number of variables. Therefore, the prominent Lauder-Paterson algorithm for computing the error linear complexity spectrum of a sequence may also be used for efficiently determining approximations of a Boolean function that depend on fewer number of variables. Moreover, it is also shown that, through this approach, low-degree approximations of a Boolean function can be also obtained in an efficient way. [ABSTRACT FROM AUTHOR]

In this paper, we first study deviations of the complete weight distribution of a linear code from that of a random code. Then, we consider a large family of subfield subcodes of algebraic-geometric codes over prime fields which include BCH codes and Goppa codes and prove that the complete weight distribution is close to that of a random code if the code length is large compared with the genus of the curve and the degree of the divisor defining the code. [ABSTRACT FROM AUTHOR]

A common way to protect data stored in DRAM and related memory systems is through the use of an error-correcting code such as the extended Hamming code. Traditionally, these error-correcting codes provide equal protection guarantees to all messages. In this paper, we focus on unequal message protection (UMP), in which a subset of messages is deemed as special, and is afforded additional error-correction protection while maintaining the same number of redundancy bits as the baseline code. UMP is a powerful approach when the special messages are chosen based on the knowledge of data patterns in context. Our objective is to construct deterministic, algebraic codes with guaranteed UMP properties, derive their cardinality bounds using novel combinatorial techniques, and to demonstrate their efficacy for realistic memory benchmarks. We first introduce a UMP alternative to the single-bit parity-check code, and then we generalize to a broader UMP code family, including a UMP alternative to the extended Hamming code, offering full double-error correction protection to special messages. Our UMP constructions, applied to main memory in high-performance computing applications, could lead to significant system-level benefits such as less frequent checkpoints in supercomputers and decreased risk of catastrophic failure from erroneous special messages. [ABSTRACT FROM AUTHOR]