An infinite family of antiprimitive cyclic codes supporting Steiner systems $S(3,8, 7^m+1)$

TL;DR

This paper constructs an infinite family of cyclic codes that support Steiner systems and 3-designs, revealing new links between coding theory and combinatorial designs with explicit parameters and symmetry properties.

Contribution

It introduces a novel infinite family of cyclic codes supporting Steiner systems and 3-designs, with detailed parameters and automorphism group analysis.

Findings

Supports Steiner systems $S(3,8, 7^m+1)$

Codes admit 3-transitive automorphism groups

Explicit parameters of the codes and designs provided

Abstract

Coding theory and combinatorial $t$-designs have close connections and interesting interplay. One of the major approaches to the construction of combinatorial t-designs is the employment of error-correcting codes. As we all known, some $t$-designs have been constructed with this approach by using certain linear codes in recent years. However, only a few infinite families of cyclic codes holding an infinite family of $3$-designs are reported in the literature. The objective of this paper is to study an infinite family of cyclic codes and determine their parameters. By the parameters of these codes and their dual, some infinite family of $3$-designs are presented and their parameters are also explicitly determined. In particular, the complements of the supports of the minimum weight codewords in the studied cyclic code form a Steiner system. Furthermore, we show that the infinite family of cyclic codes admit $3$-transitive automorphism groups.