Optimal $(2,\delta)$ Locally Repairable Codes via Punctured Simplex Codes

Locally repairable codes (LRCs) have attracted a lot of attention due to their applications in distributed storage systems. In this paper, we provide new constructions of optimal $(2, \delta)$-LRCs. Firstly, by the techniques of finite geometry, we present a sufficient condition to guarantee a punctured simplex code to be a $(2, \delta)$-LRC. Secondly, by using characteristic sums over finite fields and Krawtchouk polynomials, we construct several families of LRCs with new parameters. All of our new LRCs are optimal with respect to the generalized Cadambe-Mazumdar bound.Comment: Accepted for publication in ISIT202