New pattern erasure codes

In this paper, we study binary pattern erasure codes, i.e., binary codes that are resiliant to erasures from a family P of possible erasures. We give an algorithmic proof of the existence of a binary linear code with codewords of length n that is resiliant to erasures from P when P satisfies the properties: every pattern p ϵ P has size m and every letter in the alphabet occurs in at most c patterns. The density of the parity matrix is plays a important role in storage applications, so we also introduce a new low density code basing on graph theory.