Edge Coloring and Stopping Sets Analysis in Product Codes with MDS components

TL;DR

This paper analyzes edge coloring and stopping sets in non-binary product codes with MDS components, proposing algorithms and characterizations that improve decoding performance over erasure channels.

Contribution

It introduces a graph-based framework for analyzing stopping sets and proposes a differential evolution algorithm for optimal edge coloring in product codes.

Findings

Characterization of stopping sets up to size (d1+1)(d2+1)

Proposed coloring algorithm with low complexity per iteration

Numerical results show improved performance with double-diversity colorings

Abstract

We consider non-binary product codes with MDS components and their iterative row-column algebraic decoding on the erasure channel. Both independent and block erasures are considered in this paper. A compact graph representation is introduced on which we define double-diversity edge colorings via the rootcheck concept. An upper bound of the number of decoding iterations is given as a function of the graph size and the color palette size $M$. Stopping sets are defined in the context of MDS components and a relationship is established with the graph representation. A full characterization of these stopping sets is given up to a size $(d_1+1)(d_2+1)$, where $d_1$ and $d_2$ are the minimum Hamming distances of the column and row MDS components respectively. Then, we propose a differential evolution edge coloring algorithm that produces colorings with a large population of minimal rootcheck order symbols. The complexity of this algorithm per iteration is $o(M^{\aleph})$, for a given differential evolution parameter $\aleph$, where $M^{\aleph}$ itself is small with respect to the huge cardinality of the coloring ensemble. The performance of MDS-based product codes with and without double-diversity coloring is analyzed in presence of both block and independent erasures. In the latter case, ML and iterative decoding are proven to coincide at small channel erasure probability. Furthermore, numerical results show excellent performance in presence of unequal erasure probability due to double-diversity colorings.