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In this paper we introduce Ouroboros, a new family of Key Exchange protocols based on coding theory. The protocols propose a middle ground between the cryptosystems based on $\mathsf {QC}$ - $\mathsf {MDPC}$ codes, which feature small parameter sizes, but have a security reduction to two problems: the syndrome decoding problem and the indistinguishability of the code, and the HQC protocol, which features bigger parameters but has a security reduction to the syndrome decoding problem only. Ouroboros features a reduction to the syndrome decoding problem with only a small overhead compared to the $\mathsf {QC}$ - $\mathsf {MDPC}$ based cryptosystems. The approach is based on an ideal structure and also works for the rank metric. This yields a simple, secure and efficient approach for key exchange, the Ouroboros family of protocols. For the Hamming metric we obtain the same type of parameters (and almost the same simple decoding) as for $\mathsf {MDPC}$ based cryptosystems, but with a security reduction to decoding random quasi-cyclic codes in the Random Oracle Model. This represents a reduction of up to 38% on the public key size compared to HQC, for the most secure parameters. For the rank metric, we obtain better parameters than for RQC, saving up to 31% on the public key for the most secure set of parameters, using non homogeneous errors in Ouroboros. In this full version, the protocol and decoding algorithm have been slightly improved, additional details are given in the security proof, and the protocol is fully described for the rank metric. [ABSTRACT FROM AUTHOR]

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]

Gradient coding is a technique for straggler mitigation in distributed learning. In this paper we design novel gradient codes using tools from classical coding theory, namely, cyclic MDS codes, which compare favorably with existing solutions, both in the applicable range of parameters and in the complexity of the involved algorithms. Second, we introduce an approximate variant of the gradient coding problem, in which we settle for approximate gradient computation instead of the exact one. This approach enables graceful degradation, i.e., the $\ell _{2}$ error of the approximate gradient is a decreasing function of the number of stragglers. Our main result is that normalized adjacency matrices of expander graphs yield excellent approximate gradient codes, which enable significantly less computation compared to exact gradient coding, and guarantee faster convergence than trivial solutions under standard assumptions. We experimentally test our approach on Amazon EC2, and show that the generalization error of approximate gradient coding is very close to the full gradient while requiring significantly less computation from the workers. [ABSTRACT FROM AUTHOR]

In this paper, we develop a technique to extend any bound for the minimum distance of cyclic codes constructed from its defining sets (ds-bounds) to Abelian (or multivariate) codes through the notion of ${\mathbb B}$ -apparent distance. We also study conditions for an Abelian code to verify that its ${\mathbb B}$ -apparent distance reaches its (true) minimum distance. Then, we construct some codes as an application. [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]

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]

In this paper, the algebraic constructions of quasi-cyclic low-density parity-check (QC-LDPC) codes with the unequal error protection (UEP) property are considered. A criterion for constructing such codes via the masking technique is proposed, based on which explicit conditions on the base parity-check matrices and masking matrices to achieve UEP are provided. We also give three specific constructions of UEP QC-LDPC codes. Furthermore, a sufficient condition to ensure strict UEP is presented. Simulation results demonstrate the superiority of our constructed codes over time-sharing schemes. The constructed codes also have competitive error performance against randomly constructed equal-error-protection LDPC codes and irregular UEP LDPC codes designed based on the degree distribution. [ABSTRACT FROM PUBLISHER]

Cyclic codes are an interesting type of linear codes and have applications in communication and storage systems due to their efficient encoding and decoding algorithms. In this paper, three types of generalized cyclotomy of order two are described and three classes of cyclic codes of length n1n2 and dimension (n1n2+1)/2 are presented and analyzed, where n1 and n2 are two distinct primes. Bounds on their minimum odd-like weight are also proved. Some of the codes presented in this paper are among the best cyclic codes. [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]

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]

This paper is devoted to studying two main problems: 1) computing the apparent distance of an Abelian code and 2) giving a notion of Bose, Ray-Chaudhuri, Hocquenghem (BCH) multivariate code. To do this, we first strengthen the notion of an apparent distance by introducing the notion of a strong apparent distance; then, we present an algorithm to compute the strong apparent distance of an Abelian code, based on some manipulations of hypermatrices associated with its generating idempotent. Our method uses less computations than those given by Camion and Sabin; furthermore, in the bivariate case, the order of computation complexity is reduced from exponential to linear. Then, we use our techniques to develop a notion of a BCH code in the multivariate case, and we extend most of the classical results on cyclic BCH codes. Finally, we apply our method to the design of Abelian codes with maximum dimension with respect to a fixed apparent distance and a fixed length. [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]

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]

Let \(m\geq 3\) be an odd integer and \(p\) be an odd prime. In this paper, a number of classes of three-weight cyclic codes \(\mathcal {C}_{(1,e)}\) over \(\mathbb {F}_{p}\) , which have parity-check polynomial \(m_1(x)m_e(x)\) , are presented by examining general conditions on the parameters \(p\) , \(m\) , and \(e\) , where \(m_i(x)\) is the minimal polynomial of \(\pi ^{-i}\) over \(\mathbb {F}_{p}\) for a primitive element \(\pi \) of \(\mathbb {F}_{p^m}\) . Furthermore, for \(p\equiv 3 \pmod {4}\) and a positive integer \(e\) satisfying \((p^k+1)\cdot e\equiv 2 \pmod {p^m-1}\) for some positive integer \(k\) with \(\gcd (m,k)=1\) , the value distributions of the exponential sums \(T(a,b)=\sum \nolimits _{x\in \mathbb {F}_{p^m}}\omega ^{ \text {Tr}(ax+bx^e)}\) and \(S(a,b,c)=\sum \nolimits _{x\in \mathbb {F}_{p^m}}\omega ^{ \text {Tr}(ax+bx^e+cx^s)},\) where \(s=(p^m-1)/2\) , are determined. As an application, the value distribution of \(S(a,b,c)\) is utilized to derive the weight distribution of the cyclic codes \(\mathcal {C}_{(1,e,s)}\) with parity-check polynomial \(m_1(x)m_e(x)m_s(x)\) . In the case of \(p=3\) and even \(e\) satisfying the above condition, the dual of the cyclic code \(\mathcal {C}_{(1,e,s)}\) has optimal minimum distance. [ABSTRACT FROM PUBLISHER]

Cyclic codes have been widely used in digital communication systems and consumer electronics as they have efficient encoding and decoding algorithms. The weight distribution of cyclic codes has been an important topic of study for many years. It is in general hard to determine the weight distribution of linear codes. In this paper, a class of cyclic codes with any number of zeros is described and their weight distributions are determined. [ABSTRACT FROM AUTHOR]

Cyclic codes are a subclass of linear codes and have wide applications in consumer electronics, data storage systems, and communication systems as they have efficient encoding and decoding algorithms. In this paper, the two-prime sequence is employed to construct several classes of cyclic codes over GF(q). Lower bounds on the minimum weight of these cyclic codes are developed. Some of the codes obtained are optimal or almost optimal. The p-ranks of the twin-prime difference sets and a class of almost difference sets are computed. [ABSTRACT FROM PUBLISHER]

A new coding framework is established for channels whose outputs are overlapping pairs of symbols. Such channels are motivated by storage applications in which the spatial resolution of the reader may be insufficient to isolate adjacent symbols. Reading symbols as pairs changes the coding-theoretic error model from the standard bounded number of symbol errors to a bounded number of pair errors. Starting from the most basic coding-theoretic questions, the paper studies codes that protect against pair-errors. It provides answers on pair-error correctability conditions, code construction and decoding, and lower and upper bounds on code sizes. Asymptotic analysis of pair-error correction shows that there exist pair-error codes with rates that are strictly higher than the best known codes in the Hamming metric. [ABSTRACT FROM AUTHOR]

The guessing number of a directed graph (digraph), equivalent to the entropy of that digraph, was introduced as a direct criterion on the solvability of a network coding instance. This paper makes two contributions on the guessing number. First, we introduce an undirected graph on all possible configurations of the digraph, referred to as the guessing graph, which encapsulates the essence of dependence amongst configurations. We prove that the guessing number of a digraph is equal to the logarithm of the independence number of its guessing graph. Therefore, network coding solvability is no more a problem on the operations made by each node, but is simplified into a problem on the messages that can transit through the network. By studying the guessing graph of a given digraph, and how to combine digraphs or alphabets, we are thus able to derive bounds on the guessing number of digraphs. Second, we construct specific digraphs with high guessing numbers, yielding network coding instances where a large amount of information can transit. We first propose a construction of digraphs with finite parameters based on cyclic codes, with guessing number equal to the degree of the generator polynomial. We then construct an infinite class of digraphs with arbitrary girth for which the ratio between the linear guessing number and the number of vertices tends to one, despite these digraphs being arbitrarily sparse. These constructions yield solvable network coding instances with a relatively small number of intermediate nodes for which the node operations are known and linear, although these instances are sparse and the sources are arbitrarily far from their corresponding sinks. [ABSTRACT FROM AUTHOR]

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

A theorem of McEliece on the p-divisibility of Hamming weights in cyclic codes over Fp is generalized to Abelian codes over Z/pd Z. This work improves upon results of Helleseth—Kumar—Moreno—Shanbhag, Calderbank—Li—Poonen, Wilson, and Katz. These previous attempts are not sharp in general, i.e., do not report the full extent of the p-divisibility except in special cases, nor do they give accounts of the precise circumstances under which they do provide best possible results. This paper provides sharp results on p-divisibilities of Hamming weights and counts of any particular symbol for an arbitrary Abelian code over Z/pd Z. It also presents sharp results on 2-divisibilities of Lee and Euclidean weights for Abelian codes over Z/4Z. [ABSTRACT FROM AUTHOR]

Cyclic linear codes of block length n over a finite field Fq are linear subspaces Fqn of that are invariant under a cyclic shift of their coordinates. A family of codes is good if all the codes in the family have constant rate and constant normalized distance (distance divided by block length). It is a long-standing open problem whether there exists a good family of cyclic linear codes. A code C is r-testable if there exists a randomized algorithm which, given a word x ∈ Fqn, adaptively selects r positions, checks the entries of x in the selected positions, and makes a decision (accept or reject x) based on the positions selects and the numbers found, such that i) jf x ∈ C then x is surely accepted; ii) if dist (x, C) ⩾ ∈n then x is probably rejected. ("dist" refers to Hamming distance.) A family of codes is locally testable if all members of the family are r-testable for some constant r. This concept arose from holographic proofs/PCP's. Recently it was asked whether there exist good, locally testable families of codes. In this paper the intersection of the two questions stated is addressed. Theorem. There are no good, locally testable families of cyclic codes over any (fixed) finite field. In fact the result is stronger in that it replaces condition ii) of local testability by the condition ii) if dist (x, C) ⩾ en then x has a positive chance of being rejected. The proof involves methods from Galois theory, cyclotomy, and diophantine approximation. [ABSTRACT FROM AUTHOR]

This corrects a proposition and its proof in the paper “Sharp p-Divisibility of Weights in Abelian Codes over \BBZ/p^d\BBZ” by Daniel J. Katz (2008). This proposition was used to prove further results in the paper. The rest of the paper (including the main results and all other claims, and their proofs) stands as written, for it is not at all sensitive to the change. [ABSTRACT FROM PUBLISHER]