Infinite Families of Cyclic and Negacyclic Codes Supporting 3-Designs

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
${\mathrm {GF}}(q)$
supporting an infinite family of 3-designs and two infinite families of negacyclic codes over
${\mathrm {GF}}(q^{2})$
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
${\mathrm {GF}}(q)$
are studied. Three infinite families of almost MDS codes are also presented. A constacyclic code over
${\mathrm {GF}}(4)$
supporting a 4-design and seven open problems are also presented in this paper.