Infinite families of $2$-designs from a class of non-binary Kasami cyclic codes

TL;DR

This paper constructs infinite families of 2-designs using the supports of codewords in non-binary Kasami cyclic codes by analyzing their weight distributions, revealing new links between coding theory and combinatorial designs.

Contribution

It determines the weight distribution of certain non-binary Kasami cyclic codes and explicitly constructs infinite families of 2-designs from their codewords.

Findings

Infinite families of 2-designs derived from Kasami cyclic codes.

Explicit parameters of the constructed 2-designs.

New connections between non-binary cyclic codes and combinatorial designs.

Abstract

Combinatorial $t$-designs have been an important research subject for many years, as they have wide applications in coding theory, cryptography, communications and statistics. The interplay between coding theory and $t$-designs has been attracted a lot of attention for both directions. It is well known that a linear code over any finite field can be derived from the incidence matrix of a $t$-design, meanwhile, that the supports of all codewords with a fixed weight in a code also may hold a $t$-design. In this paper, by determining the weight distribution of a class of linear codes derived from non-binary Kasami cyclic codes, we obtain infinite families of $2$-designs from the supports of all codewords with a fixed weight in these codes, and calculate their parameters explicitly.