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

Xiaoni Du; Rong Wang; Chunming Tang; Qi Wang

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arXiv · 1903.08153

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

Xiaoni Du·Rong Wang·Chunming Tang·Qi Wang

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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 $m=4$.

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Xiaoni Du, Rong Wang, Chunming Tang, Qi Wang. 2019-03-19. Infinite families of $2$-designs from two classes of binary cyclic codes with three nonzeros. https://arxiv.org/abs/1903.08153

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