Locally Repairable Codes: Joint Sequential–Parallel Repair for Multiple Node Failures.

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 $r$. [ABSTRACT FROM AUTHOR]