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

Xiaoni Du; Rong Wang; Chunming Tang; Qi Wang

doi:10.3934/amc.2020106

Article Contents

2022, Volume 16, Issue 1: 157-168. Doi: 10.3934/amc.2020106

This issue Previous Article Orbit codes from forms on vector spaces over a finite field Next Article Optimal antiblocking systems of information sets for the binary codes related to triangular graphs

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

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 and Information, China West Normal University, Nanchong, Sichuan 637002, China
4.
Department of Computer Science and Engineering, Southern University of Science and Technology, Shenzhen, Guangdong 518055, China

^* Corresponding author: Rong Wang
^* Corresponding author: Rong Wang

Received: May 2020

Early access: August 2020

Published: February 2022

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

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

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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 $

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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 $

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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 $

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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) $

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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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