Infinite families of 2-designs from two classes of binary cyclic codes with three nonzeros

Xiaoni Du, Rong Wang, Chunming Tang, Qi Wang

Abstract: Combinatorial $ t $-designs have been an interesting topic in combinatorics for decades. It is a basic fact that the codewords of a fixed weight in a code may hold a $ t $-design. Till now only a small amount of work on constructing $ t $-designs from codes has been done. In this paper, we determine the weight distributions of two classes of cyclic codes: one related to the triple-error correcting binary BCH codes, and the other related to the cyclic codes with parameters satisfying the generalized Kasami case, respectively. We then obtain infinite families of $ 2 $-designs from these codes by proving that they are both affine-invariant codes, and explicitly determine their parameters. In particular, the codes derived from the dual of binary BCH codes hold five $ 3 $-designs when $ m = 4 $.