Infinite families of 2-designs and 3-designs from linear codes

Abstract

The interplay between coding theory and $t$-designs started many years ago. While every $t$-design yields a linear code over every finite field, the largest $t$ for which an infinite family of $t$-designs is derived directly from a linear or nonlinear code is $t=3$. Sporadic $4$-designs and $5$-designs were derived from some linear codes of certain parameters. The major objective of this paper is to construct many infinite families of $2$-designs and $3$-designs from linear codes. The parameters of some known $t$-designs are also derived. In addition, many conjectured infinite families of $2$-designs are also presented.