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A new family of codes, called clustering-correcting codes, is presented in this paper. This family of codes is motivated by the special structure of the data that is stored in DNA-based storage systems. The data stored in these systems has the form of unordered sequences, also called strands, and every strand is synthesized thousands to millions of times, where some of these copies are read back during sequencing. Due to the unordered structure of the strands, an important task in the decoding process is to place them in their correct order. This is usually accomplished by allocating part of the strand for an index. However, in the presence of errors in the index field, important information on the order of the strands may be lost. Clustering-correcting codes ensure that if the distance between the index fields of two strands is small, their data fields have large distance. It is shown how this property enables to place the strands together in their correct clusters even in the presence of errors. We present lower and upper bounds on the size of clustering-correcting codes and an explicit construction of these codes which uses only a single symbol of redundancy. The results are first presented for the Hamming metric and are then extended for the edit distance. [ABSTRACT FROM AUTHOR]

Constacyclic codes are well-known generalizations of cyclic and negacyclic codes. Due to their rich algebraic structure, constacyclic codes are used to construct quantum codes and symbol-pair codes. Let ${\mathbb {F}}_{q}$ be a finite field with order $q$ , where $q$ is a positive power of a prime $p$. Suppose that $n$ is a positive integer and the product of distinct prime factors of $n$ divides $q-1$ , i.e., $rad(n)\mid (q-1)$. In this paper, we explicitly factorize the polynomial $x^{n}-\lambda $ for each $\lambda \in {\mathbb {F}}_{q}^{*}$. As applications, first, we obtain all repeated-root $\lambda $ -constacyclic codes and their dual codes of length $np^{s}$ over ${\mathbb {F}}_{q}$ ; second, we determine all simple-root LCD cyclic codes and LCD negacyclic codes of length $n$ over ${\mathbb {F}}_{q}$ ; third, we list all self-dual repeated-root negacyclic codes of length $np^{s}$ over ${\mathbb {F}}_{q}$. In contrast to known results, the lengths of constacyclic codes in this paper have more flexible parameters. [ABSTRACT FROM AUTHOR]

In an interesting paper, Prof. C. Ding provided three constructions of cyclic codes of length being a product of two primes. Numerical data shows that many codes from these constructions are best cyclic codes of the same length and dimension over the same finite field. However, not much is known about these codes. In this paper, we explain some of the numerical data by developing a general method on cyclic codes of composite length and on estimating the minimum distance. We also provide a general construction of cyclic codes of composite length which are related to Ding’s constructions. Numerical data shows that it produces many best cyclic codes as well. Finally, we point out how these cyclic codes can be used to construct convolutional codes with large free distance. [ABSTRACT FROM AUTHOR]

Constant dimension codes (CDCs), as special subspace codes, have received a lot of attention due to their application in random network coding. This paper introduces a family of new codes, called rank metric codes with given ranks (GRMCs), to generalize the parallel construction in [Xu and Chen, IEEE Trans. Inf. Theory, 64 (2018), 6315–6319] and the classic multilevel construction. A Singleton-like upper bound and a lower bound for GRMCs derived from Gabidulin codes are given. Via GRMCs, two effective constructions for CDCs are presented by combining the parallel construction and the multilevel construction. Many CDCs with larger size than the previously best known codes are given. The ratio between the new lower bound and the known upper bound for $(4\delta,2\delta,2\delta)_{q}$ -CDCs is calculated. It is greater than 0.99926 for any prime power $q$ and any $\delta \geq 3$. [ABSTRACT FROM AUTHOR]

Limited magnitude asymmetric error model is well suited for flash memory. In this paper, we consider the construction of asymmetric codes correcting single error over $\mathbb {Z}_{2^{k}r}$ which is based on so called $B_{1}[{4}](2^{k}r)$ set. In fact, we reduce the construction of a maximal size $B_{1}[{4}](2^{k}r)$ set for $k\geq 3$ to the construction of a maximal size $B_{1}[{4}](2^{k-3}r)$ set. Finally, we give an explicit formula of a maximal size $B_{1}[{4}](4r)$ set and some lower bounds of a maximal size $B_{1}[{4}](2r)$ set. By computer searching up to $q\leq 106$ , we conjecture that those lower bounds are tight. [ABSTRACT FROM AUTHOR]

In this paper, we construct several classes of maximum distance separable (MDS) codes via generalized Reed-Solomon (GRS) codes and extended GRS codes, where we can determine the dimensions of their Euclidean hulls or Hermitian hulls. It turns out that the dimensions of Euclidean hulls or Hermitian hulls of the codes in our constructions can take all or almost all possible values. As a consequence, we can apply our results to entanglement-assisted quantum error-correcting codes (EAQECCs) and obtain several new families of MDS EAQECCs with flexible parameters. The required number of maximally entangled states of these MDS EAQECCs can take all or almost all possible values. Moreover, several new classes of q-ary MDS EAQECCs of length $ {n} > {q}+1$ are also obtained. [ABSTRACT FROM AUTHOR]

Let $q$ be a prime power and $m\geq 3$ be odd. Suppose that $n=\frac {q^{2m}-1}{a}$ with $a|(q^{m}+1)$ and $3\leq a \leq 2(q^{2}-q+1)$. This paper mainly determines the actual maximum designed distance of Hermitian dual-containing Bose-Chaudhuri-Hocquenghem (BCH) codes over $\mathbb {F}_{q^{2}}$ of length $n$. Firstly, we give the maximum designed distance $\delta _{m,a}^{R}$ of narrow-sense Hermitian dual-containing BCH codes. Secondly, we show that there are also non-narrow-sense ones of designed distance up to $\delta _{m,a}^{R}$. It is worth mentioning that our maximum designed distance $\delta _{m,a}^{R}>\lceil \frac {a}{2}\rceil \delta _{m}^{A}$ , where $\delta _{m}^{A}$ is given by Aly et al. (IEEE Trans. Inf. Theory, vol. 53, no. 3, pp. 1183-1188, 2007). Thus, many families of Hermitian dual-containing BCH codes with relatively large designed distance are obtained. Using the Hermitian construction to them, we can subsequently construct different classes of nonprimitive quantum codes, which are new in the sense that their parameters are not covered in the literature. [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]

Recently, Etzion et al. introduced metrics on ${\mathbb{F}}_{2}^{n}$ based on directed graphs on $n$ vertices and developed some basic coding theory on directed graph metric spaces. In this paper, we consider the problem of classifying directed graphs, which admit the extended Hamming codes to be a perfect code. We first consider weighted poset metrics as a natural generalization of poset metrics and investigate interrelation between the weighted poset metrics and the directed graph-based metrics. In the next, we classify weighted posets on a set with eight elements and directed graphs on eight vertices, which admit the extended Hamming code $\tilde {\mathcal {H }}_{3}$ to be a two-perfect code. We also construct some families of such structures for any $k \geq 3$ , which can be viewed as generalizations of some results presented by Etzion et al. and Hyun and Kim. Those families enable us to construct packing or covering codes of radius 2 under certain maps. [ABSTRACT FROM AUTHOR]

In group testing, simple binary-output tests are designed to identify a small number $t$ of defective items that are present in a large population of $N$ items. Each test takes as input a group of items and produces a binary output indicating whether the group is free of the defective items or contains one or more of them. In this paper, we study a relaxation of the combinatorial group testing problem. A matrix is called $(t,\epsilon )$ -disjunct if it gives rise to a nonadaptive group testing scheme with the property of identifying a uniformly random $t$ -set of defective subjects out of a population of size $N$ with false positive probability of an item at most $\epsilon $ . We establish a new connection between $(t,\epsilon )$ -disjunct matrices and error correcting codes based on the dual distance of the codes and derive estimates of the parameters of codes that give rise to such schemes. Our methods rely on the moments of the distance distribution of codes and inequalities for moments of sums of independent random variables. We also provide a new connection between group testing schemes and combinatorial designs. [ABSTRACT FROM PUBLISHER]

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]

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]

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]

A q -ary code of length n , size M , and minimum distance d code. An (n,q^{k},n-k+1)_{q} code is called a maximum distance separable (MDS) code. In this paper, some MDS codes over small alphabets are classified. It is shown that every (k+d-1,q^{k},d)_{q} code with k\geq 3 , d \geq 3 , is equivalent to a linear code with the same parameters. This implies that the (6,5^4,3)5 code and the (n,7^{n-2},3)7 MDS codes for n\in \6,7,8\ are unique. The classification of one-error-correcting 8-ary MDS codes is also finished; there are 14, 8, 4, and 4 equivalence classes of (n,8^n-2,3)8 codes for $n=6,7,8$ , and 9, respectively. One of the equivalence classes of perfect (9,8^{7},3)_{8} codes corresponds to the Hamming code and the other three are nonlinear codes for which there exists no previously known construction. [ABSTRACT FROM PUBLISHER]

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]