Locally repairable codes have attracted lots of interest in Distributed Storage Systems. If a symbol of a code can be repaired respectively by t disjoint groups of other symbols, each groups has size at most r, we say that the code symbol has (r, t)-locality. In this paper, we employ parity-check matrix to construct information single-parity (r, t)-locality LRCs. All our codes attain the Singleton-like bound of LRCs where each repair group contains a single parity symbol and thus are optimal.

One of the main problems in list decoding is to determine the tradeoff between the list decoding radius and the rate of the codes w.r.t. a given metric. In this paper, we first describe a “stronger-weaker” relationship between two distinct metrics of the same code, then we show that the list decodability of the stronger metric can be deduced from the weaker metric directly. In particular, when we focus on matrix codes, we can obtain list decodability of matrix code w.r.t. the cover metric from the Hamming metric and the rank metric. Moreover, we deduce a Johnson-like bound of the list decoding radius for cover metric codes, which improved the result of [20]. In addition, the condition for a metric that whether the list decoding radius w.r.t. this metric and the rate are bounded by the Singleton bound is presented.

In this paper, we describe the Galois dual of rank metric codes in the ambient space FQn×m and FQmn, where Q=qe. We obtain connections between the duality of rank metric codes with respect to distinct Galois inner products. Furthermore, for 0 ≤ s < e, we introduce the concept of qsm-dual bases of FQm over FQ and obtain some conditions about the existence of qsm-self-dual basis.

The Galois hull of linear code is defined to be the intersection of the code and its Galois dual. In this paper, we investigate the Galois hulls of cyclic codes over Fqr. For any integer s≤r, we present some sufficient and necessary conditions that cyclic codes have l-dimensional s-Galois hull. Moreover, we prove that a cyclic code C has l-dimensional s-Galois hull iff C has l-dimensional (r-s)-Galois hull. In particular, we also present the sufficient and necessary condition for cyclic codes with 1-dimensional Galois hulls and the relationship between cyclic codes with 1-dimensional Galois hulls and cyclic codes with Galois complementary duals. Some optimal cyclic codes with Galois hulls are obtained. Finally, we explicitly construct a class of cyclic codes with 1-Galois linear complementary dual over Fq3.