Infinite families of $2$-designs from a class of linear codes related to Dembowski-Ostrom functions

TL;DR

This paper constructs a new class of linear codes from cyclic codes related to Dembowski-Ostrom functions, determines their weight distribution, and derives infinite families of 2-designs with explicit parameters from these codes.

Contribution

It introduces a novel class of linear codes based on Dembowski-Ostrom functions and explicitly constructs infinite families of 2-designs from these codes.

Findings

Determined the weight distribution of the constructed codes.

Established infinite families of 2-designs from code supports.

Calculated parameters of the 2-designs explicitly.

Abstract

Due to their important applications to coding theory, cryptography, communications and statistics, combinatorial $t$-designs have been attracted lots of research interest for decades. The interplay between coding theory and $t$-designs has on going for many years. As we all known, $t$-designs can be used to derive linear codes over any finite field, as well as the supports of all codewords with a fixed weight in a code also may hold a $t$-design. In this paper, we first construct a class of linear codes from cyclic codes related to Dembowski-Ostrom functions. By using exponential sums, we then determine the weight distribution of the linear codes. Finally, we obtain infinite families of $2$-designs from the supports of all codewords with a fixed weight in these codes. Furthermore, the parameters of $2$-designs are calculated explicitly.