Infinite families of cyclic and negacyclic codes supporting 3-designs

TL;DR

This paper constructs new infinite families of cyclic and negacyclic codes over finite fields that support 3-designs, including the first known negacyclic codes with this property, and explores their parameters and related codes.

Contribution

It introduces the first infinite families of negacyclic codes supporting 3-designs, expanding the known connections between coding theory and combinatorial designs.

Findings

Constructed infinite families of cyclic codes supporting 3-designs.

Constructed two infinite families of negacyclic codes supporting 3-designs.

Determined parameters and weight distributions of these codes.

Abstract

Interplay between coding theory and combinatorial $t$-designs has been a hot topic for many years for combinatorialists and coding theorists. Some infinite families of cyclic codes supporting infinite families of $3$-designs have been constructed in the past 50 years. However, no infinite family of negacyclic codes supporting an infinite family of $3$-designs has been reported in the literature. This is the main motivation of this paper. Let $q=p^m$, where $p$ is an odd prime and $m \geq 2$ is an integer. The objective of this paper is to present an infinite family of cyclic codes over $\gf(q)$ supporting an infinite family of $3$-designs and two infinite families of negacyclic codes over $\gf(q^2)$ supporting two infinite families of $3$-designs. The parameters and the weight distributions of these codes are determined. The subfield subcodes of these negacyclic codes over $\gf(q)$ are studied. Three infinite families of almost MDS codes are also presented. A constacyclic code over GF($4$) supporting a $4$-design and six open problems are also presented in this paper.