Locally repairable codes (LRC) have been studied from two approaches to locally repair multiple failed nodes: 1) parallel approach, in which a coordinate $i$ of an $[n,k,d]$ linear code is said to have locality $r$ and availability $t$ if there exist $t$ disjoint repair sets each of which contains at most $r$ other coordinates that can recover the value of the $i$ -th coordinate; 2) sequential approach, in which the erased symbols (failed nodes) are repaired, one by one, and any previously repaired node can be used to repair the remaining failed nodes. In this paper, we first consider LRC aiming at joint sequential-parallel repairing multiple failed nodes, and study the $(n,k,r,t,u)$ -ELRCs (Exact locally repairable codes) which are $[n,k]$ linear codes with the property that any set of failed nodes of size at most $t$ can be simultaneously repaired in parallel mode, and each element of a set $E$ of failed nodes of size at most $u$ can be sequentially repaired by $r$ ( $r< k$ ) other coordinates. We present a method by which with a given parity-check matrix of an $(n,k,r,t,u)$ -ELRC with minimum Hamming distance $d$ , a new ELRC with minimum Hamming distance $2d$ and availability $t+1$ is constructed that can repair each set of failed nodes $E$ of size at most $2u+1$ in sequential mode and this repair is done in at most $u-t+2$ steps. We construct a big family of LRCs by making use of orthogonal Latin rectangles and permutation cubes and some other combinatorial designs; the constructed codes contain the family of direct product codes; we also use $m$ -dimensional permutation cubes to construct LRCs with short block length for each $r$ .

中文翻译：

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本地可修复代码：多节点故障的联合顺序并行修复

局部修复代码 (LRC) 已经从两种方法进行了研究，以局部修复多个故障节点：1) 并行方法，其中一个坐标 $i$ 的 $[n,k,d]$ 线性代码被称为具有局部性 $r$ 和可用性 $t$ 如果存在 $t$ 不相交的修复集，每个修复集最多包含 $r$ 其他可以恢复值的坐标 $i$ -th坐标；2）顺序方法，其中被擦除的符号（故障节点）被一个一个地修复，任何先前修复的节点都可以用于修复剩余的故障节点。在本文中，我们首先考虑 LRC 旨在联合顺序并行修复多个故障节点，并研究 $(n,k,r,t,u)$ - ELRC（精确的本地可修复代码） $[n,k]$ 具有任何一组最大大小的失败节点的特性的线性代码 $t$ 可以并行方式同时修复，一个集合的每个元素 $E$ 最大大小的失败节点数 $u$ 可以依次修复 $r$ ( $r<k$ ) 其他坐标。我们提出了一种方法，通过它具有给定的奇偶校验矩阵 $(n,k,r,t,u)$ -具有最小汉明距离的ELRC $d$ , 具有最小汉明距离的新 ELRC $2d$ 和可用性 $t+1$ 构造可以修复每组故障节点 $E$ 最多大小 $2u+1$ 在顺序模式下，此修复最多在 $u-t+2$ 脚步。我们通过使用正交拉丁矩形和置换立方体以及其他一些组合设计构建了一个大的 LRC 家族；构造的代码包含直接产品代码族；我们也用 百万美元 维置换立方体来构造每个块长度较短的 LRC $r$ .