Binary Linear-Time Erasure Decoding for Non-Binary LDPC codes

TL;DR

This paper presents a linear-time binary erasure decoding algorithm for non-binary LDPC codes using an extended binary representation, simplifying decoding complexity while providing insights into code structure and applications.

Contribution

Introduces an extended binary representation and a linear-time erasure decoding algorithm for non-binary LDPC codes, enhancing decoding efficiency and understanding of code structure.

Findings

Decoding complexity depends linearly on alphabet size

Extended binary representation corresponds to a covering graph

Insights into stopping set structures for non-binary LDPC codes

Abstract

In this paper, we first introduce the extended binary representation of non-binary codes, which corresponds to a covering graph of the bipartite graph associated with the non-binary code. Then we show that non-binary codewords correspond to binary codewords of the extended representation that further satisfy some simplex-constraint: that is, bits lying over the same symbol-node of the non-binary graph must form a codeword of a simplex code. Applied to the binary erasure channel, this description leads to a binary erasure decoding algorithm of non-binary LDPC codes, whose complexity depends linearly on the cardinality of the alphabet. We also give insights into the structure of stopping sets for non-binary LDPC codes, and discuss several aspects related to upper-layer FEC applications.