The subfield codes and subfield subcodes of a family of MDS codes

TL;DR

This paper investigates subfield subcodes of a classical family of MDS codes over GF(2^m), constructing quaternary cyclic codes that are often optimal and deriving infinite 3-designs from them.

Contribution

It introduces new quaternary cyclic codes as subfield subcodes of known MDS codes and explores their optimality and combinatorial design properties.

Findings

Quaternary cyclic codes are distance-optimal in some cases.

Infinite families of 3-designs are constructed from these codes.

The codes are very good in general, extending the utility of the original MDS codes.

Abstract

Maximum distance separable (MDS) codes are very important in both theory and practice. There is a classical construction of a family of $[2^m+1, 2u-1, 2^m-2u+3]$ MDS codes for $1 \leq u \leq 2^{m-1}$, which are cyclic, reversible and BCH codes over $\mathrm{GF}(2^m)$. The objective of this paper is to study the quaternary subfield subcodes and quaternary subfield codes of a subfamily of the MDS codes for even $m$. A family of quaternary cyclic codes is obtained. These quaternary codes are distance-optimal in some cases and very good in general. Furthermore, infinite families of $3$-designs from these quaternary codes are presented.