A New Bound on the Minimum Distance of Cyclic Codes Using Small-Minimum-Distance Cyclic Codes

TL;DR

This paper introduces a new bound on the minimum distance of q-ary cyclic codes, improving existing bounds like the Hartmann--Tzeng bound, and provides methods for efficient decoding and conditions for low-distance codes.

Contribution

It proposes a novel bound based on small-minimum-distance cyclic codes, explicitly relates it to BCH and HT bounds, and refines it for specific code families, enabling better decoding strategies.

Findings

The new bound often surpasses the Hartmann--Tzeng bound.

Efficient decoding up to the new bound is possible using the Extended Euclidean Algorithm.

Conditions for binary cyclic codes with minimum distance two or three are established.

Abstract

A new bound on the minimum distance of q-ary cyclic codes is proposed. It is based on the description by another cyclic code with small minimum distance. The connection to the BCH bound and the Hartmann--Tzeng (HT) bound is formulated explicitly. We show that for many cases our approach improves the HT bound. Furthermore, we refine our bound for several families of cyclic codes. We define syndromes and formulate a Key Equation that allows an efficient decoding up to our bound with the Extended Euclidean Algorithm. It turns out that lowest-code-rate cyclic codes with small minimum distances are useful for our approach. Therefore, we give a sufficient condition for binary cyclic codes of arbitrary length to have minimum distance two or three and lowest code-rate