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

• * Corresponding author: Rong Wang

Xiaoni Du's research is supported by NSFC grant No. 61772022. Chunming Tang's research is supported by NSFC Grant No. 11871058. Qi Wang's research is supported by NSFC Grant No. 61672015

• 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 $m = 4$.

Mathematics Subject Classification: Primary: 05B05, 94B05, 11T23, 11T71.

 Citation:

• Table 1.  The weight distribution of ${\overline{{\mathcal{C}_1}^{\bot}}}^{\bot}$

 Weight Multiplicity $0$ $1$ $2^{2s-1}$ $29\times2^{6s-5}-33\times2^{4s-5}+17\times2^{2s-3}-2$ $2^{2s-1}-2^{s-1}$ $\frac{2}{15}\times2^{2s}(3\times2^{4s}+5\times2^{2s}-8)$ $2^{2s-1}+2^{s-1}$ $\frac{2}{15}\times2^{2s}(3\times2^{4s}+5\times2^{2s}-8)$ $2^{2s-1}-2^s$ $\frac{7}{3}\times2^{4s-4}(2^{2s}-1)$ $2^{2s-1}+2^s$ $\frac{7}{3}\times2^{4s-4}(2^{2s}-1)$ $2^{2s-1}-2^{s+1}$ $\frac{1}{15}\times2^{2s-4}(2^{4s-2}-5\times2^{2s-2}+1)$ $2^{2s-1}+2^{s+1}$ $\frac{1}{15}\times2^{2s-4}(2^{4s-2}-5\times2^{2s-2}+1)$ $2^{2s}$ $1$

Table 2.  The weight distribution of ${\overline{{\mathcal{C}_2}^{\bot}}}^{\bot}$ when $d' = d$

 Weight Multiplicity $0$ $1$ $2^{2s-1}-2^{s-1}$ $2^{2s}(2^s-1)(2^{2(s+d)}-2^{2s+d}-2^{2s}+2^{s+2d}-2^{s+d}+2^{2d})/(2^{2d}-1)$ $2^{2s-1}+2^{s-1}$ $2^{2s}(2^s-1)(2^{2(s+d)}-2^{2s+d}-2^{2s}+2^{s+2d}-2^{s+d}+2^{2d})/(2^{2d}-1)$ $2^{2s-1}-2^{s+d-1}$ $2^{2(s-d)}(2^{s+d}-1)(2^{2s}-1)/(2^{2d}-1)$ $2^{2s-1}+2^{s+d-1}$ $2^{2(s-d)}(2^{s+d}-1)(2^{2s}-1)/(2^{2d}-1)$ $2^{2s-1}$ $2(2^{3s-d}-2^{2(s-d)}+1)(2^{2s}-1)$ $2^{2s}$ $1$

Table 3.  The weight distribution of ${\overline{{\mathcal{C}_2}^{\bot}}}^{\bot}$ when $d' = 2d$

 Weight Multiplicity $0$ $1$ $2^{2s-1}-2^{s-1}$ $2^{2s+3d}(2^s-1)(2^{2s}-2^{2(s-d)}-2^{2s-3d}+2^s-2^{s-d}+1)/(2^{2d}-1)(2^d+1)$ $2^{2s-1}+2^{s-1}$ $2^{2s+3d}(2^s-1)(2^{2s}-2^{2(s-d)}-2^{2s-3d}+2^s-2^{s-d}+1)/(2^{2d}-1)(2^d+1)$ $2^{2s-1}-2^{s+d-1}$ $2^{2s-d}(2^{2s}-1)(2^s+2^{s-d}+2^{s-2d}+1)/(2^d+1)^2$ $2^{2s-1}+2^{s+d-1}$ $2^{2s-d}(2^{2s}-1)(2^s+2^{s-d}+2^{s-2d}+1)/(2^d+1)^2$ $2^{2s-1}$ $2(2^{2s}-1)(2^{3s-d}-2^{3s-2d}+2^{3s-3d}-2^{3s-4d}+2^{3s-5d}+2^{2s-d}-2^{2s-2d+1} +2^{2s-3d}-2^{2s-4d}+1)$ $2^{2s-1}-2^{s+2d-1}$ $2^{2s-4d}(2^{s-d}-1)(2^{2s}-1)/(2^d+1)(2^{2d}-1)$ $2^{2s-1}+2^{s+2d-1}$ $2^{2s-4d}(2^{s-d}-1)(2^{2s}-1)/(2^d+1)(2^{2d}-1)$ $2^{2s}$ $1$

Table 4.  The weight distribution of ${\mathcal{C}_2}$ when $d' = d$

 Weight Multiplicity $0$ $1$ $2^{2s-1}-2^{s-1}$ $2^{s-1}(2^{2s}-1)(2^{2(s+d)}-2^{2s+d}-2^{2s}+2^{s+2d}-2^{s+d}+2^{2d})/(2^{2d}-1)$ $2^{2s-1}+2^{s-1}$ $2^{s-1}(2^s-1)^2(2^{2(s+d)}-2^{2s+d}-2^{2s}+2^{s+2d}-2^{s+d}+2^{2d})/(2^{2d}-1)$ $2^{2s-1}-2^{s+d-1}$ $2^{s-d-1}(2^{s+d}-1)(2^{2s}-1)(2^{s-d}+1)/(2^{2d}-1)$ $2^{2s-1}+2^{s+d-1}$ $2^{s-d-1}(2^{s+d}-1)(2^{2s}-1)(2^{s-d}-1)/(2^{2d}-1)$ $2^{2s-1}$ $(2^{3s-d}-2^{2(s-d)}+1)(2^{2s}-1)$

Table 5.  The weight distribution of ${\mathcal{C}_2}$ when $d' = 2d$

 Weight Multiplicity $0$ $1$ $2^{2s-1}-2^{s-1}$ $\frac{2^{s+3d-1}(2^{2s}-1)(2^{2s}-2^{2(s-d)}-2^{2s-3d}+2^s-2^{s-d}+1)}{(2^{2d}-1)(2^d+1)}$ $2^{2s-1}+2^{s-1}$ $\frac{2^{2s+3d-1}(2^s-1)^2(2^{2s}-2^{2(s-d)}-2^{2s-3d}+2^s-2^{s-d}+1)}{(2^{2d}-1)(2^d+1)}$ $2^{2s-1}-2^{s+d-1}$ $2^{s-1}(2^{2s}-1)(2^s+2^{s-d}+2^{s-2d}+1)(2^{s-d}+1)/(2^d+1)^2$ $2^{2s-1}+2^{s+d-1}$ $2^{s-1}(2^{2s}-1)(2^s+2^{s-d}+2^{s-2d}+1)(2^{s-d}-1)/(2^d+1)^2$ $2^{2s-1}$ $(2^{2s}-1)(2^{3s-d}-2^{3s-2d}+2^{3s-3d}-2^{3s-4d}+2^{3s-5d} +2^{2s-d}-2^{2s-2d+1}+2^{2s-3d}-2^{2s-4d}+1)$ $2^{2s-1}-2^{s+2d-1}$ $2^{s-2d-1}(2^{s-d}-1)(2^{2s}-1)(2^{s-2d}+1)/(2^d+1)(2^{2d}-1)$ $2^{2s-1}+2^{s+2d-1}$ $2^{s-2d-1}(2^{s-d}-1)(2^{2s}-1)(2^{s-2d}-1)/(2^d+1)(2^{2d}-1)$
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