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

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 $m=4$.

arXiv admin note: text overlap with arXiv:1903.07459

Cited by in corpus (4)

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• Combinatorial t-designs from quadratic functions
• Infinite families of $2$-designs from a class of linear codes related to Dembowski-Ostrom functions
• Linear Codes Of 2-Designs As Subcodes Of The Extended Generalized Reed-Muller Codes