Infinite families of 2-designs from two classes of binary cyclic codes with three nonzeros (Q2070264)

Combinatorial \(t\)-designs have been investigated for many decades. It is well known that the codewords of a fixed weight in a code may hold a \(t\)-design. In this paper, the authors 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. Infinite families of 2-designs are obtained 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.