Locally Repairable Codes Based on Permutation Cubes and Latin Squares

The importance of Locally Repairable Codes (LRCs) lies on their applications in distributed storage systems (DSSs). There are two approaches to repair multiple failed nodes: the parallel approach, in which a set of failed nodes are simultaneously repaired; and the sequential method, wherein the failed nodes are repaired successively by making use of known nodes, including those already repaired. LRCs in the joint sequential-parallel mode were investigated with the aim of reducing the repair time in the sequential mode, and in this study, we continue the investigation by providing LRCs with higher repair tolerance. We propose a construction of generator matrices for binary LRCs based on back-circulant Latin squares and
$t$
-dimensional permutation cubes. The codes based on back-circulant Latin squares have locality
$r=5$
and availability
$t \geq 4 $
in the parallel mode and, in the case where
$t+1$
is neither even nor a multiple of 3, they possess a short block length compared to their counterparts [Tamo and Barg (2014) and Wang and Zhang (2015)]. We present LRCs based on
$t$
-dimensional permutation
$m$
-cubes with block length
$m^{t}$
, locality
$r=2m-3$
, and availability
$t \geq 3$
in parallel mode. These codes can repair any set of failed nodes of size up to
$2^{t}-1$
in the sequential mode in at most
$t-1$
steps. It is shown that these codes are
overall local
with repair tolerance
$2t$
in the parallel mode. Finally, we introduce the LRC-AL class of codes, a new class of LRCs which satisfies the property that for any pair of nodes
$i,j$
, there is a repair set for the failed node
$i$
that contains the live node
$j$
.