Decoding of Repeated-Root Cyclic Codes up to New Bounds on Their Minimum Distance

TL;DR

This paper extends classical bounds on minimum distance to repeated-root cyclic codes and introduces new decoding algorithms for burst errors, enhancing error correction capabilities.

Contribution

It generalizes existing bounds to repeated-root cyclic codes and proposes new syndrome-based and probabilistic decoding algorithms.

Findings

Generalized bounds for repeated-root cyclic codes

Developed a syndrome-based burst error decoding algorithm

Outlined a probabilistic decoding procedure with quadratic complexity

Abstract

The well-known approach of Bose, Ray-Chaudhuri and Hocquenghem and its generalization by Hartmann and Tzeng are lower bounds on the minimum distance of simple-root cyclic codes. We generalize these two bounds to the case of repeated-root cyclic codes and present a syndrome-based burst error decoding algorithm with guaranteed decoding radius based on an associated folded cyclic code. Furthermore, we present a third technique for bounding the minimum Hamming distance based on the embedding of a given repeated-root cyclic code into a repeated-root cyclic product code. A second quadratic-time probabilistic burst error decoding procedure based on the third bound is outlined. Index Terms Bound on the minimum distance, burst error, efficient decoding, folded code, repeated-root cyclic code, repeated-root cyclic product code