Codes, Differentially $\delta$ -Uniform Functions, and $t$ -Designs

An Infinite Family of Linear Codes Supporting 4-Designs

“…There are three sets of sufficient conditions under which the incidence structure (P(C), B k (C)) is a t-design for some positive integer t. The first set of conditions is described in the Assmus-Mattson Theorem [1]. The second set of conditions is documented in a generalised Assmus-Mattson Theorem [20]. The third set of conditions is in terms of the automorphism group of the code C [12, p. 308].…”

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

“…Very recently, Liu et al [28] obtained some 3-transitive automorphism groups from a class of BCH codes and derived some combinatorial 3-designs with this way. Another major way to construct t-designs with linear codes is the use of the Assmus-Mattson Theorem (AM Theorem for short) in [10,Theorem 4.14] and the generalized version of the AM Theorem in [22], which was recently employed to construct a number of t-designs (see, for example, [10], [24], [25]). The following theorem is a generalized version of the AM Theorem, which was developed in [22] and will be needed in this paper.…”

“…Theorem 1. [22] Let C be a linear code over the finite field GF(q) with length ν and minimum distance d. Let C ⊥ denote the dual of C with minimum distance d ⊥ . Let s and t be two positive integers such that t < min{d, d ⊥ }.…”

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

“…Very recently, Tang and Ding [22] settled this long‐standing problem by presenting an infinite family of BCH codes holding an infinite family of 4‐ designs. More constructions of ‐designs can be found in [10,17,21,20,23,24,27] and related references.…”