Asymptotically MDS Array BP-XOR Codes

TL;DR

This paper introduces asymptotically MDS array BP-XOR codes that simplify code construction while maintaining low decoding complexity, using a discrete geometry-based method to approach the MDS property asymptotically.

Contribution

It proposes a new class of asymptotically MDS array BP-XOR codes that ease the construction process compared to exact MDS codes, with low decoding complexity.

Findings

The proposed codes approach MDS properties asymptotically.

A simple discrete geometry-based construction method is provided.

The codes maintain low complexity decoding using belief propagation.

Abstract

Belief propagation or message passing on binary erasure channels (BEC) is a low complexity decoding algorithm that allows the recovery of message symbols based on bipartite graph prunning process. Recently, array XOR codes have attracted attention for storage systems due to their burst error recovery performance and easy arithmetic based on Exclusive OR (XOR)-only logic operations. Array BP-XOR codes are a subclass of array XOR codes that can be decoded using BP under BEC. Requiring the capability of BP-decodability in addition to Maximum Distance Separability (MDS) constraint on the code construction process is observed to put an upper bound on the maximum achievable code block length, which leads to the code construction process to become a harder problem. In this study, we introduce asymptotically MDS array BP-XOR codes that are alternative to exact MDS array BP-XOR codes to pave the way for easier code constructions while keeping the decoding complexity low with an asymptotically vanishing coding overhead. We finally provide and analyze a simple code construction method that is based on discrete geometry to fulfill the requirements of the class of asymptotically MDS array BP-XOR codes.