Permutation Decoding and the Stopping Redundancy Hierarchy of Linear Block Codes

TL;DR

This paper explores the stopping redundancy hierarchy of linear block codes and introduces new decoding strategies combining permutation decoding and iterative message passing to improve performance over the binary erasure channel.

Contribution

It develops a theoretical framework linking stopping redundancy and permutation decoding, and proposes novel decoding methods utilizing automorphisms and s-SAD sets.

Findings

New bounds on stopping redundancy hierarchy

Decoding strategies combining permutation and iterative decoding

Introduction of s-SAD sets for erasure correction

Abstract

We investigate the stopping redundancy hierarchy of linear block codes and its connection to permutation decoding techniques. An element in the ordered list of stopping redundancy values represents the smallest number of possibly linearly dependent rows in any parity-check matrix of a code that avoids stopping sets of a given size. Redundant parity-check equations can be shown to have a similar effect on decoding performance as permuting the coordinates of the received codeword according to a selected set of automorphisms of the code. Based on this finding we develop new decoding strategies for data transmission over the binary erasure channel that combine iterative message passing and permutation decoding in order to avoid errors confined to stopping sets. We also introduce the notion of s-SAD sets, containing the smallest number of automorphisms of a code with the property that they move any set of not more than s erasures into positions that do not correspond to stopping sets within a judiciously chosen parity-check matrix.