Edge coloring and stopping sets analysis in product codes with MDS components

Fanny Jardel, Joseph Boutros

Research output: Contribution to journalArticle

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. 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) , for a given differential evolution parameter, where M itself is small with respect to the huge cardinality of the coloring ensemble. Stopping sets of a product code 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)2, where d is the minimum Hamming distance of the MDS component code. The performance of MDS-based product codes with and without double-diversity coloring is analyzed in presence of both the block and the 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.

Original languageEnglish
Article number7819551
Pages (from-to)1439-1462
Number of pages24
JournalIEEE Transactions on Information Theory
Volume63
Issue number3
DOIs
Publication statusPublished - 1 Mar 2017

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Keywords

  • Codes on graphs
  • Differential evolution
  • Distributed storage
  • Diversity
  • Edge coloring
  • Erasure channel
  • Iterative decoding
  • MDS codes
  • Product codes
  • Stopping sets

ASJC Scopus subject areas

  • Information Systems
  • Computer Science Applications
  • Library and Information Sciences

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