Systematic Memory MDS Sliding Window Codes Over Erasure Channels

Abstract

Memory maximum-distance-separable (mMDS) sliding window codes are a type of erasure codes with high erasure-correction capability and low decoding delay. In this paper, we study two types of systematic mMDS sliding window codes over erasure channels, <italic>i.e.,</italic> scalar codes defined over a finite field <inline-formula> <tex-math notation= LaTeX >$GF(2^{L})$ </tex-math></inline-formula>, and vector codes defined over a vector space <inline-formula> <tex-math notation= LaTeX >$GF(2)^{L}$ </tex-math></inline-formula>. We first devise an efficient heuristic algorithm to produce an mMDS sliding window scalar code over relatively small <inline-formula> <tex-math notation= LaTeX >$GF(2^{L})$ </tex-math></inline-formula>. Then, we investigate a special class of mMDS sliding window vector codes whose encoding/decoding are achieved by basic circular-shift and bit-wise XOR operations, and propose a general method to generate such mMDS vector codes. Our complexity analysis shows that the proposed vector codes yield much lower encoding/decoding complexity than the scalar codes. The theoretical and numerical results also demonstrate that mMDS sliding window codes dominate MDS block codes in terms of decoding delay and erasure-correction capability.