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Locally repairable codes (LRCs) are a class of codes designed for the local correction of erasures. They have received considerable attention in recent years due to their applications in distributed storage. Most existing results on LRCs do not explicitly take into consideration the field size $q$ , i.e., the size of the code alphabet. In particular, for the binary case, only a few results are known. In this paper, we present an upper bound on the minimum distance $d$ of linear LRCs with availability, based on the work of Cadambe and Mazumdar. The bound takes into account the code length $n$ , dimension $k$ , locality $r$ , availability $t$ , and field size $q$ . Then, we study the binary linear LRCs in three aspects. First, we focus on analyzing the locality of some classical codes, i.e., cyclic codes and Reed–Muller codes, and their modified versions, which are obtained by applying the operations of extend, shorten, expurgate, augment, and lengthen. Next, we construct LRCs using phantom parity-check symbols and multi-level tensor product structure, respectively. Compared with other previous constructions of binary LRCs with fixed locality or minimum distance, our construction is much more flexible in terms of code parameters, and gives various families of high-rate LRCs, some of which are shown to be optimal with respect to their minimum distance. Finally, the availability of LRCs is studied. We investigate the locality and availability properties of several classes of one-step majority-logic decodable codes, including cyclic simplex codes, cyclic difference-set codes, and 4-cycle free regular low-density parity-check codes. We also show the construction of a long LRC with availability from a short one-step majority-logic decodable code. [ABSTRACT FROM PUBLISHER]

Cyclic codes with a few weights are very useful in the design of frequency hopping sequences and the development of secret sharing schemes. In this paper, we mainly use Gauss sums to represent the Hamming weights of cyclic codes whose duals have two zeroes. A lower bound of the minimum Hamming distance is determined. In some cases, we give the Hamming weight distributions of the cyclic codes. In particular, we obtain a class of three-weight optimal cyclic codes achieving the Griesmer bound, which generalizes a Vega’s result, and several classes of cyclic codes with a few weights, which solve an open problem proposed by Vega. [ABSTRACT FROM AUTHOR]

The generalized Hamming weights (GHWs) of linear codes are fundamental parameters, the knowledge of which is of great interest in many applications. However, to determine the GHWs of linear codes is difficult in general. In this paper, we study the GHWs for a family of reducible cyclic codes and obtain the complete weight hierarchy in several cases. This is achieved by extending the idea of Yang et al. into higher dimension and by employing some interesting combinatorial arguments. It shall be noted that these cyclic codes may have arbitrary number of nonzeros. [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]

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]

In this paper, we extend two earlier works further in two directions and compute the weight distribution of these cyclic codes under more relaxed conditions. It is interesting to note that many cyclic codes in the family are optimal and have only a few non-zero weights. Besides using similar ideas, we carry out some subtle manipulation of certain exponential sums. [ABSTRACT FROM AUTHOR]

The generalized Hamming weights dr(C) of a linear code C are a natural generalization of the minimum Hamming distance d(C)[=d_{1}(C)] and have become an important research object in coding theory since Wei’s originary work in 1991. In this paper, two general formulas on dr(C) for irreducible cyclic codes are presented using Gauss sums and the weight hierarchy \{d_{1}(C), d_{2}(C), \ldots , d_{k}(C)\}~(k=\dim C)$ is completely determined for several cases. [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]

Cyclic codes are an interesting class of linear codes due to their efficient encoding and decoding algorithms. Bose-Ray-Chaudhuri-Hocquenghem (BCH) codes form a subclass of cyclic codes and are very important in both theory and practice as they have good error-correcting capability and are widely used in communication systems, storage devices, and consumer electronics. However, the dimension and minimum distance of BCH codes are not known in general. The objective of this paper is to determine the Bose and minimum distances of a class of narrow-sense primitive BCH codes. [ABSTRACT FROM AUTHOR]

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]

Recently, there has been intensive research on the weight distributions of cyclic codes. In this paper, we compute the weight distributions of three classes of cyclic codes with Niho exponents. More specifically, we obtain two classes of binary three-weight and four-weight cyclic codes and a class of nonbinary four-weight cyclic codes. The weight distributions follow from the determination of value distributions of certain exponential sums. Several examples are presented to show that some of our codes are optimal and some have the best known parameters. [ABSTRACT FROM AUTHOR]

Cyclic codes are an interesting type of linear codes and have wide applications in communication and storage systems due to their efficient encoding and decoding algorithms. In this paper, let \(\Bbb F_r\) be a finite field with \(r=q^m\) . Suppose that \(g_1, g_2 \in \Bbb F_r^*\) are not conjugates over \(\Bbb F_q\) , \( \operatorname {ord}(g_1)=n_1\) , \( \operatorname {ord}(g_2)=n_2\) , \(d=\gcd (n_1, n_2)\) , and \(n={n_1n_2}/ d\) . Let \(\Bbb F_q(g_1)=\Bbb F_{q^{m_1}}, \Bbb F_q(g_2)=\Bbb F_{q^{m_2}}\) , and \(\text{T}_i\) denote the trace function from \(\Bbb F_{q^{m_i}}\) to \(\Bbb F_q\) for \(i=1, 2\) . We define a cyclic code \(\mathcal C_{(q, m, n_1, n_2)}=\{c(a,b) : a \in \Bbb F_{q^{m_1}}, b \in \Bbb F_{q^{m_2}}\},\) where \(c(a,b)=(\text {T}_1(ag_1^0)+\text {T}_2(bg_2^0), \text {T}_1(ag_1^1)+\text {T}_2(bg_2^1), \ldots , \text {T}_1(ag_1^{n-1})+\text {T}_2(bg_2^{n-1})).\) We mainly use Gauss periods to present the weight distribution of the cyclic code \(\mathcal C_{(q, m, n_1, n_2)}\) . As applications, we determine the weight distribution of cyclic code \(\mathcal C_{(q, m,q^{m_1}-1,q^{m_2}-1)}\) with \(\gcd (m_1,m_2)=1\) ; in particular, it is a three-weight cyclic code if \(\gcd (q-1,m_1-m_2)=1\) . We also explicitly determine the weight distributions of some classes of cyclic codes including several classes of four-weight cyclic codes. [ABSTRACT FROM AUTHOR]