Binary Codes with Locality for Four Erasures

In this paper, codes with locality for four erasures are considered. An upper bound on the rate of codes with locality with sequential recovery from four erasures is derived. The rate bound derived here is field independent. An optimal construction for binary codes meeting this rate bound is also provided. The construction is based on regular graphs of girth $6$ and employs the sequential approach of locally recovering from multiple erasures. An extension of this construction that generates codes which can sequentially recover from five erasures is also presented.