Generalizing Bounds on the Minimum Distance of Cyclic Codes Using Cyclic Product Codes

TL;DR

This paper introduces two new bounds on the minimum distance of q-ary cyclic codes by embedding them into cyclic product codes, enabling improved decoding strategies and generalizing existing bounds.

Contribution

It proposes two generalized bounds on cyclic code minimum distances using cyclic product code embeddings, with the first allowing unique decoding and the second providing a stronger bound.

Findings

First bound enables decoding up to the new minimum distance

Second bound is stronger and more comprehensive

Embedding technique generalizes other bounds

Abstract

Two generalizations of the Hartmann--Tzeng (HT) bound on the minimum distance of q-ary cyclic codes are proposed. The first one is proven by embedding the given cyclic code into a cyclic product code. Furthermore, we show that unique decoding up to this bound is always possible and outline a quadratic-time syndrome-based error decoding algorithm. The second bound is stronger and the proof is more involved. Our technique of embedding the code into a cyclic product code can be applied to other bounds, too and therefore generalizes them.