Optimal Constructions of LRCs based on Cayley Table

Network coding techniques have emerged as an essential tool in the aspect of storage applications, especially in distributed storage systems (DSS). Locally repairable code (LRC) is one of these. LRCs with availability are designed to tolerate single node failure in a DSS while still allowing the stored data to be recovered. A code is called information symbol (n,k,r,t) LRC if each information coordinate can be achieved by at least t disjoint repair sets, containing at most r other coordinates. This paper includes some explicit constructions through the standard parity check matrix of a class of information symbol (n, k, r, t) LRCs based on two different Cayley tables of a finite field. The minimum distance of these constructed codes is optimal against the bound $d \leq n - k - \left\lceil {\frac{{kt}}{n}} \right\rceil + t + 1$ in which some of these codes have non trivial optimal minimum distance also.