An Infinite Family of Steiner Systems from Cyclic Codes

Abstract

Steiner systems are a fascinating topic of combinatorics. The most studied Steiner systems are S(2, 3, v) (Steiner triple systems), S(3, 4, v) (Steiner quadruple systems), and S(2, 4, v). There are a few infinite families of Steiner systems S(2, 4, v) in the literature. The objective of this paper is to present an infinite family of Steiner systems S(2, 4, 2 m ) for all m ≡ 2 (mod 4) ≥ 6 from cyclic codes. This may be the first coding-theoretic construction of an infinite family of Steiner systems S (2, 4, v). As a by-product, many infinite families of 2-designs are also reported in this paper.

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

“…By definition, such design (P (C ), B w (C )) could have some repeated blocks, or could be simple, or may be trivial. In this way, many t-designs have been constructed from linear codes (see, for example, [10], [10], [14], [15], [19], [21], [23]). A major way to construct combinatorial t-designs with linear codes over finite fields is the use of linear codes with t-transitive or t-homogeneous automorphism groups (see [10,Theorem 4.18]) and some combinatorial t-designs (see, for example, [7]) were obtained by this way.…”

“…Meanwhile, a linear code C may induce a t-design under certain conditions. As far as we know, a lot of 2-designs and 3-designs have been constructed from some special linear codes (see, for example, [10], [14], [15], [19], [21]). Recently, an infinity family of linear codes holding 4-designs was settled by Tang and Ding in [23].…”

Infinite families of 2‐designs derived from affine‐invariant codes

“…When and is a subset of , the designs from the code and their relatives have been studied in [7,6,9,8,13], where . Besides, when is odd, the cyclic codes considered in [11,12,27] are the special cases of .…”

“…In 2017 and 2018, infinite families of 2‐designs and 3‐designs were obtained from several different classes of linear codes by Ding [6], Ding and Li [9] by applying the Assmus–Mattson Theorem. Besides, Ding [7] presented an infinite family of Steiner systems by a class of affine‐invariant codes. Recently, Du et al have derived infinite families of 2‐designs from several classes of affine‐invariant codes [11,13,12].…”

Combinatorial t-designs from special functions

“…. , i t ) ⊥ become the affine-invariant binary codes treated in [12], where a class of Steiner systems S(2, 4, 2 m ) was obtained. Notice that these codes were treated as extended cyclic codes in [12].…”