Permutation Decoding and the Stopping Redundancy Hierarchy of Cyclic and Extended Cyclic Codes

TL;DR

This paper introduces the stopping redundancy hierarchy for linear codes, derives bounds using probabilistic methods, and develops a novel automorphism group decoding technique that approaches maximum likelihood performance.

Contribution

It defines the stopping redundancy hierarchy, derives bounds for it, and proposes a new decoding method combining permutation and iterative decoding.

Findings

Bounds on stopping redundancy hierarchy established

Automorphism group decoding performs close to maximum likelihood

New bounds on permutation requirements for erasure correction

Abstract

We introduce the notion of the stopping redundancy hierarchy of a linear block code as a measure of the trade-off between performance and complexity of iterative decoding for the binary erasure channel. We derive lower and upper bounds for the stopping redundancy hierarchy via Lovasz's Local Lemma and Bonferroni-type inequalities, and specialize them for codes with cyclic parity-check matrices. Based on the observed properties of parity-check matrices with good stopping redundancy characteristics, we develop a novel decoding technique, termed automorphism group decoding, that combines iterative message passing and permutation decoding. We also present bounds on the smallest number of permutations of an automorphism group decoder needed to correct any set of erasures up to a prescribed size. Simulation results demonstrate that for a large number of algebraic codes, the performance of the new decoding method is close to that of maximum likelihood decoding.