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

Rong Wang; Xiaoni Du; Cuiling Fan

doi:10.3934/amc.2020088

Article Contents

2021, Volume 15, Issue 4: 663-676. Doi: 10.3934/amc.2020088

This issue Previous Article Correlation distribution of a sequence family generalizing some sequences of Trachtenberg Next Article Information set decoding in the Lee metric with applications to cryptography

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

1.
College of Mathematics and Statistics, Northwest Normal University, Lanzhou, Gansu 730070, China
2.
Guangxi Key Laboratory of Cryptography and Information Security, Guilin University of Electronic Technology, Guilin, Guangxi 541004, China
3.
School of Mathematics, Southwest Jiaotong University, Chengdu, Sichuan 610000, China

^* Corresponding author: Xiaoni Du
^* Corresponding author: Xiaoni Du

Received: November 30, 2019

Revised: January 31, 2020

Early access: June 2020

Published: November 2021

The second author is supported by NSFC grant No. 61772022. The third author is supported by NSFC grant No. 11971395

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Abstract

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.

Keywords:

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

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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 $

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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 $

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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 $

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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 $

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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 $

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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 $

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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 $

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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 $

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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 $

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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 $

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