Some $3$-designs and shortened codes from binary cyclic codes with three zeros

TL;DR

This paper explores the construction of 3-designs from extended and shortened binary cyclic codes with three zeros, expanding the known families of codes associated with combinatorial designs and identifying some optimal codes.

Contribution

It introduces new infinite families of 3-designs derived from extended and shortened cyclic codes with three zeros, including explicit parameters and optimal code examples.

Findings

Extended codes of binary cyclic codes with three zeros hold 3-designs.

Shortened codes from these cyclic codes are explicitly characterized.

Some shortened codes are optimal or nearly optimal.

Abstract

Linear codes and $t$-designs are interactive with each other. It is well known that some $t$-designs have been constructed by using certain linear codes in recent years. However, only a small number of infinite families of the extended codes of linear codes holding an infinite family of $t$-designs with $t\geq 3$ are reported in the literature. In this paper, we study the extended codes of the augmented codes of a class of binary cyclic codes with three zeros and their dual codes, and show that those codes hold $3$-designs. Furthermore, we obtain some shortened codes from the studied cyclic codes and explicitly determine their parameters. Some of those shortened codes are optimal or almost optimal.