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# Infinite families of 2-designs from a class of non-binary Kasami cyclic codes

• * Corresponding author: Xiaoni Du
The second author is supported by NSFC grant No. 61772022. The third author is supported by NSFC grant No. 11971395
• Combinatorial $t$-designs have been an important research subject for many years, as they have wide applications in coding theory, cryptography, communications and statistics. The interplay between coding theory and $t$-designs has been attracted a lot of attention for both directions. It is well known that a linear code over any finite field can be derived from the incidence matrix of a $t$-design, meanwhile, that the supports of all codewords with a fixed weight in a code also may hold a $t$-design. In this paper, by determining the weight distribution of a class of linear codes derived from non-binary Kasami cyclic codes, we obtain infinite families of $2$-designs from the supports of all codewords with a fixed weight in these codes, and calculate their parameters explicitly.

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

 Citation: • • Table 1.  The weight distribution of ${\overline{{\mathbb{C}}^{\bot}}}^{\bot}$ when $d' = d$ is odd

 Weight Multiplicity $0$ $1$ $(p-1)(p^{m-1}-p^{s-1})$ $\frac{1}{2}p^{m+d}(p^s+1)(p^m-1)/(p^d+1)$ $p^{m-1}(p-1)+p^{s-1}$ $\frac{1}{2}p^{m+d}(p-1)(p^s+1)(p^m-1)/(p^d+1)$ $(p-1)(p^{m-1}+p^{s-1})$ $\frac{p^{m+d}(p^m-2p^{m-d}+1)(p^s-1)}{2(p^d-1)}$ $p^{m-1}(p-1)-p^{s-1}$ $\frac{p^{m+d}(p-1)(p^m-2p^{m-d}+1)(p^s-1)}{2(p^d-1)}$ $p^{m-1}(p-1)\pm (-1)^{\frac{p-1}{2}}p^{s+\frac{d-1}{2}}$ $\frac{1}{2}p^{3s-2d}(p-1)(p^m-1)$ $p^{s+d-1}(p-1)(p^{s-d}+1)$ $p^{m-2d}(p^{s-d}-1)(p^m-1)/(p^{2d}-1)$ $p^{m-1}(p-1)-p^{s+d-1}$ $\frac{p^{m-2d}(p-1)(p^{s-d}-1)(p^m-1)}{(p^{2d}-1)}$ $p^{m-1}(p-1)$ $p(p^{3s-d}-p^{3s-2d}+p^{3s-2d-1}+p^{3s-3d}$ $-p^{m-2d}+1)(p^m-1)$ $p^m$ $p-1$

Table 2.  The weight distribution of ${\overline{{\mathbb{C}}^{\bot}}}^{\bot}$ when $d' = d$ is even

 Weight Multiplicity $0$ $1$ $p^{s-1}(p-1)(p^s-1)$ $\frac{1}{2}p^{m+d}(p^s+1)(p^m-1)/(p^d+1)$ $p^{s-1}(p^{s+1}-p^s+1)$ $\frac{1}{2}p^{m+d}(p-1)(p^s+1)(p^m-1)/(p^d+1)$ $p^{s-1}(p-1)(p^s+1)$ $p^{m+d}(p^m-2p^{m-d}+1)(p^s-1)/2(p^d-1)$ $p^{s-1}(p^{s+1}-p^s-1)$ $p^{m+d}(p-1)(p^m-2p^{m-d}+1)(p^s-1)/2(p^d-1)$ $p^{s+\frac{d}{2}-1}(p-1)(p^{s-\frac{d}{2}} \pm 1)$ $\frac{1}{2}p^{3s-2d}(p^m-1)$ $p^{s+\frac{d}{2}-1}(p^{s-\frac{d}{2}+1}-p^{s-\frac{d}{2}}\pm 1)$ $\frac{1}{2}p^{3s-2d}(p-1)(p^m-1)$ $p^{s+d-1}(p-1)(p^{s-d}+1)$ $p^{m-2d}(p^{s-d}-1)(p^m-1)/(p^{2d}-1)$ $p^{s+d-1}(p^{s-d+1}-p^{s-d}-1)$ $\frac{p^{m-2d}(p-1)(p^{s-d}-1)(p^m-1)}{(p^{2d}-1)}$ $p^{m-1}(p-1)$ $p(p^{3s-d}-p^{3s-2d}+p^{3s-3d}-p^{m-2d}$ $+1)(p^m-1)$ $p^m$ $p-1$

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

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

Table 4.  The value of $T(a,b,c,h)$ when $d' = d$ is odd ($j\in \mathbb{F}_p$)

 Value Corresponding Condition $p^{m-1}+\varepsilon p^{s-1}(p-1)$ $S(a,b,c)=\varepsilon p^s\zeta_p^j\; \mathrm{and}\; h+j=0$ $p^{m-1}-\varepsilon p^{s-1}$ $S(a,b,c)=\varepsilon p^s\zeta_p^j\; \mathrm{and}\; h+j\neq0$ $0$ $S(a,b,c)= p^m\; \mathrm{and}\; h\neq0$ $p^{m-1}\pm(-1)^{\frac{p-1}{2}}p^{s+\frac{d-1}{2}}$ $S(a,b,c)=\varepsilon \sqrt{p^*}p^{s+\frac{d-1}{2}}\zeta_p^j\; \mathrm{and}\; \eta'(h+j)=\pm\varepsilon$ $p^{m-1}-p^{s+d-1}(p-1)$ $S(a,b,c)=-p^{s+d}\zeta_p^j \; \mathrm{and}\; h+j=0$ $p^{m-1}+p^{s+d-1}$ $S(a,b,c)$ $=-p^{s+d}\zeta_p^j \; \mathrm{and}\; h+j\neq0$ $p^{m-1}$ $S(a,b,c)=0\; \mathrm{or}\; S(a,b,c)=\varepsilon\sqrt{p^*}\zeta_p^jp^{s+\frac{d-1}{2}}\; \mathrm{and}$ $h+j = 0$ $p^m$ $S(a,b,c)=p^m\; \mathrm{and}\; h = 0$

Table 5.  The weight distribution of ${\overline{{\mathbb{C}}^{\bot}}}^{\bot}$ when $d' = d$ is odd

 Value Multiplicity $(p-1)(p^{m-1}-p^{s-1})$ $M_1+(p-1)M_3$ $p^{m-1}(p-1)+p^{s-1}$ $(p-1)M_1+(p-1)^2M_3$ $(p-1)(p^{m-1}+p^{s-1})$ $M_2+(p-1)M_4$ $p^{m-1}(p-1)-p^{s-1}$ $(p-1)M_2+(p-1)^2M_4$ $p^{m-1}(p-1)\pm(-1)^{\frac{p-1}{2}}p^{s+\frac{d-1}{2}}$ $(p-1)M_5+\frac{(p-1)^2}{2}(M_{6,1}+M_{6,-1})$ $p^{s+d-1}(p-1)(p^{s-d}+1)$ $M_7+(p-1)M_8$ $p^{m-1}(p-1)-p^{s+d-1}$ $(p-1)M_7+(p-1)^2M_8$ $p^{m-1}(p-1)$ $pM_9+2M_5+(p-1)(M_{6,1}+M_{6,-1})$ $p^m$ $p-1$

Table 6.  The weight distribution of ${\overline{{\mathbb{C}}^{\bot}}}^{\bot}$ when $d' = d$ is even

 Value Multiplicity $p^{s-1}(p-1)(p^s-1)$ $M_1+(p-1)M_3$ $p^{s-1}(p^{s+1}-p^s+1)$ $(p-1)M_1+(p-1)^2M_3$ $p^{s-1}(p-1)(p^s+1)$ $M_2+(p-1)M_4$ $p^{s-1}(p^{s+1}-p^s-1)$ $(p-1)M_2+(p-1)^2M_4$ $p^{s+\frac{d}{2}-1}(p^{s-\frac{d}{2}+1}-p^{s-\frac{d}{2}}\pm1)$ $(p-1)M_{5,\pm1}+(p-1)^2M_{6,\pm1}$ $p^{s+\frac{d}{2}-1}(p-1)(p^{s-\frac{d}{2}}\pm1)$ $M_{5,\mp1}+(p-1)M_{6,\mp1}$ $p^{s+d-1}(p-1)(p^{s-d}+1)$ $M_7+(p-1)M_8$ $p^{s+d-1}(p^{s-d+1}-p^{s-d}-1)$ $(p-1)M_7+(p-1)^2M_8$ $p^{m-1}(p-1)$ $pM_9$ $p^m$ $p-1$

Table 7.  The weight distribution of ${\overline{{\mathbb{C}}^{\bot}}}^{\bot}$ when $d' = 2d$

 Value Multiplicity $p^{s-1}(p-1)(p^s+1)$ $M_1+(p-1)M_2$ $p^{s-1}(p^{s+1}-p^s-1)$ $(p-1)M_1+(p-1)^2M_2$ $p^{s+d-1}(p-1)(p^{s-d}-1)$ $M_3+(p-1)M_4$ $p^{s+d-1}(p^{s-d+1}-p^{s-d}+1)$ $(p-1)M_3+(p-1)^2M_4$ $p^{s+2d-1}(p-1)(p^{s-2d}+1)$ $M_5+(p-1)M_6$ $p^{s+2d-1}(p^{s-2d+1}-p^{s-2d}-1)$ $(p-1)M_5+(p-1)^2M_6$ $p^{m-1}(p-1)$ $pM_7$ $p^m$ $p-1$

Table 8.  The value distribution of $S(a,b,c)$ when $d' = d$ is odd

 Value Multiplicity $p^s$ $M_1= \frac{1}{2}p^{s+d-1}(p^s+1)(p^s+p-1)(p^m-1)/(p^d+1)$ $-p^s$ $M_2=\frac{1}{2}p^{s+d-1}(p^s-1)(p^s-p+1)(p^m-2p^{m-d}+1)/(p^d-1)$ $\zeta^j_pp^s$ $M_3=\frac{1}{2}p^{s+d-1}(p^m-1)^2/(p^d+1)$ $-\zeta^j_pp^s$ $M_4=\frac{1}{2}p^{s+d-1}(p^m-2p^{m-d}+1)(p^m-1)/(p^d-1)$ $\varepsilon\sqrt{p^*}p^{s+\frac{d-1}{2}}$ $M_5=\frac{1}{2}p^{3s-2d-1}(p^m-1)$ $\varepsilon\sqrt{p^*}p^{s+\frac{d-1}{2}}\zeta^j_p$ $M_{6,\varepsilon}=\frac{1}{2}p^{m-\frac{3d+1}{2}}(p^{s-\frac{d+1}{2}}+\varepsilon\eta'(-j))(p^m-1)$ $-p^{s+d}$ $M_7=p^{s-d-1}(p^{s-d}-1)(p^{s-d}-p+1)(p^m-1)/(p^{2d}-1)$ $-p^{s+d}\zeta^j_p$ $M_8= p^{s-d-1}(p^{m-2d}-1)(p^m-1)/(p^{2d}-1)$ $0$ $M_9=(p^{3s-d}-p^{3s-2d}+p^{3s-3d}-p^{m-2d}+1)(p^m-1)$ $p^m$ $1$

Table 9.  The value distribution of $S(a,b,c)$ when $d' = d$ is even

 Value Multiplicity $p^s$ $M_1= \frac{1}{2}p^{s+d-1}(p^s+1)(p^s+p-1)(p^m-1)/(p^d+1)$ $-p^s$ $M_2=\frac{1}{2}p^{s+d-1}(p^s-1)(p^s-p+1)(p^m-2p^{m-d}+1)/(p^d-1)$ $\zeta^j_pp^s$ $M_3=\frac{1}{2}p^{s+d-1}(p^m-1)^2/(p^d+1)$ $-\zeta^j_pp^s$ $M_4=\frac{1}{2}p^{s+d-1}(p^m-2p^{m-d}+1)(p^m-1)/(p^d-1)$ $\varepsilon p^{s+\frac{d}{2}}$ $M_{5,\varepsilon}=\frac{1}{2}p^{m-\frac{3d}{2}-1}(p^{s-\frac{d}{2}}+\varepsilon(p-1))(p^m-1)$ $\varepsilon p^{s+\frac{d}{2}}\zeta^j_p$ $M_{6,\varepsilon}=\frac{1}{2}p^{m-\frac{3d}{2}-1}(p^{s-\frac{d}{2}}-\varepsilon)(p^m-1)$ $-p^{s+d}$ $M_7=p^{s-d-1}(p^{s-d}-1)(p^{s-d}-p+1)(p^m-1)/(p^{2d}-1)$ $-p^{s+d}\zeta^j_p$ $M_8= p^{s-d-1}(p^{s-d}-1)(p^{s-d}+1)(p^m-1)/(p^{2d}-1)$ $0$ $M_9=(p^{3s-d}-p^{3s-2d}+p^{3s-3d}-p^{m-2d}+1)(p^m-1)$ $p^m$ $1$

Table 10.  The value distribution of $S(a,b,c)$ when $d' = 2d$

 Value Multiplicity $-p^s$ $M_1=\frac{p^{s+3d-1}(p^s-1)(p^s-p+1)(p^m-p^{m-2d}-p^{m-3d}+p^s-p^{s-d}+1)}{(p^d+1)(p^{2d}-1)}$ $-\zeta^j_pp^s$ $M_2=\frac{p^{s+3d-1}(p^m-p^{m-2d}-p^{m-3d}+p^s-p^{s-d}+1)(p^m-1)}{(p^d+1)(p^{2d}-1)}$ $p^{s+d}$ $M_3=\frac{p^{s-1}(p^{s-d}+p-1)(p^s+p^{s-d}+p^{s-2d}+1)(p^m-1)}{(p^d+1)^2}$ $p^{s+d}\zeta^j_p$ $M_4=\frac{p^{s-1}(p^{s-d}-1)(p^s+p^{s-d}+p^{s-2d}+1)(p^m-1)}{(p^d+1)^2}$ $-p^{s+2d}$ $M_5=\frac{p^{s-2d-1}(p^{s-d}-1)(p^{s-2d}-p+1)(p^m-1)}{(p^d+1)(p^{2d}-1)}$ $-p^{s+2d}\zeta^j_p$ $M_6=\frac{p^{s-2d-1}(p^{s-d}-1)(p^{s-2d}+1)(p^m-1)}{(p^d+1)(p^{2d}-1)}$ $0$ $M_7=(p^{3s-d}-p^{3s-2d}+p^{3s-3d}-p^{3s-4d}+p^{3s-5d}+p^{m-d}-$ $2p^{m-2d}+p^{m-3d}-p^{m-4d}+1)(p^m-1)$ $p^m$ $1$
• Tables(10)

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