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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]

In this paper, a generic construction of codebooks based on binary codes is introduced. With this generic construction, a few previous constructions of optimal codebooks are extended, and a new class of codebooks almost meeting the Levenshtein bound is presented. Exponentially many codebooks meeting or almost meeting the Levenshtein bound from binary codes are obtained in this paper. The codebooks constructed in this paper have alphabet size 4. As a byproduct, three bounds on the parameters of binary codes are derived. [ABSTRACT FROM PUBLISHER]

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]

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

Let p be a prime, and \lambda be a nonzero element of the finite field \mathbb Fp^{m} . The \lambda -constacyclic codes of length p^s over \mathbb Fp^{m} are linearly ordered under set-theoretic inclusion, i.e., they are the ideals \langle (x-\lambda 0)^{i} \rangle , 0 \leq i \leq p^s of the chain ring [(\mathbb Fp^{m[x]})/({\langle x^{p^{s}}-\lambda \rangle })] . This structure is used to establish the symbol-pair distances of all such $\lambda$ -constacyclic codes. Among others, all maximum distance separable symbol-pair constacyclic codes of length p^s are obtained. [ABSTRACT FROM AUTHOR]

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]

We consider the following multiplication-based tests to check if a given function $f: \mathbb {F}_{q}^{n}\to \mathbb {F} _{q}$ is a codeword of the Reed–Muller code of dimension $n$ and order $d$ over the finite field $\mathbb {F}_{q}$ for prime $q$ (i.e., $f$ is the evaluation of a degree- $d$ polynomial over $\mathbb {F}_{q}$ for $q$ prime). $Test_{e,k}$ : pick $P_{1},\ldots ,P_{k}$ independent random degree- $e$ polynomials and accept if the function $fP_{1}\cdots P_{k}$ is the evaluation of a degree- $(d+ek)$ polynomial (i.e., is a codeword of the Reed–Muller code of dimension $n$ and order $(d+ek)$). We prove the robust soundness of the above-mentioned tests for large values of $e$ , answering a question of Dinur and Guruswami. Previous soundness analyses of these tests were known only for the case when either $e=1$ or $k=1$. Even for the case $k=1$ and $e>1$ , earlier soundness analyses were not robust. We also analyze a derandomized version of this test, where (for example) the polynomials $P_{1},\ldots ,P_{k}$ can be the same random polynomial $P$. This generalizes a result of Guruswami et al. One of the key ingredients that go into the proof of this robust soundness is an extension of the standard Schwartz–Zippel lemma over general finite fields $\mathbb {F}_{q}$ , which may be of independent interest. [ABSTRACT FROM AUTHOR]