Decoding Cyclic Codes up to a New Bound on the Minimum Distance

TL;DR

This paper introduces a new lower bound on the minimum distance of q-ary cyclic codes, improving existing bounds and providing a quadratic-time decoding algorithm based on advanced algebraic techniques.

Contribution

It proposes a novel bound that surpasses the BCH and HT bounds for certain codes and develops an efficient decoding algorithm utilizing the Euclidean Algorithm and a generalized Forney's formula.

Findings

New bound improves upon BCH and HT bounds for some codes

Decoding algorithm operates in quadratic time

Error location and evaluation methods are generalized and optimized

Abstract

A new lower bound on the minimum distance of q-ary cyclic codes is proposed. This bound improves upon the Bose-Chaudhuri-Hocquenghem (BCH) bound and, for some codes, upon the Hartmann-Tzeng (HT) bound. Several Boston bounds are special cases of our bound. For some classes of codes the bound on the minimum distance is refined. Furthermore, a quadratic-time decoding algorithm up to this new bound is developed. The determination of the error locations is based on the Euclidean Algorithm and a modified Chien search. The error evaluation is done by solving a generalization of Forney's formula.