    doi: 10.3934/amc.2020106

## 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

Received  May 2020 Published  August 2020

Fund Project: 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 $m = 4$.

Citation: Xiaoni Du, Rong Wang, Chunming Tang, Qi Wang. Infinite families of 2-designs from two classes of binary cyclic codes with three nonzeros. Advances in Mathematics of Communications, doi: 10.3934/amc.2020106
##### References:
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##### References:
  E. F. Assmus, Jr. and J. D. Key, Designs and Their Codes, volume 103 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1992.  doi: 10.1017/CBO9781316529836.  Google Scholar  E. F. Assmus, Jr. and H. F. Mattson, Jr, New $5$-designs, J. Combinatorial Theory, 6 (1969), 122-151.  doi: 10.1016/S0021-9800(69)80115-8.  Google Scholar  T. Beth, D. Jungnickel and H. Lenz, Design Theory, Cambridge University Press, Cambridge, 1986. Google Scholar  C. J. Colbourn and R. Mathon, Steiner systems, In Handbook of Combinatorial Designs, Second Edition, pages 128–135, Chapman and Hall/CRC, 2006. https://www.researchgate.net/publication/329786723. Google Scholar  C. Ding, Codes from Difference Sets, World Scientific, 2015. Google Scholar  C. Ding, Designs from Linear Codes, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2019. doi: 10.1142/11101.  Google Scholar  C. Ding, Infinite families of $3$-designs from a type of five-weight code, Des. Codes Cryptogr., 86 (2018), 703-719.  doi: 10.1007/s10623-017-0352-6.  Google Scholar  C. Ding and C. Li, Infinite families of $2$-designs and $3$-designs from linear codes, Discrete Math., 340 (2017), 2415-2431.  doi: 10.1016/j.disc.2017.05.013.  Google Scholar  W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511807077.  Google Scholar  T. Kasami, S. Lin and W. W. Peterson, Some results on cyclic codes which are invariant under the affine group and their applications, Information and Control, 11 1967,475–496. doi: 10.1016/S0019-9958(67)90691-2.  Google Scholar  G. T. Kennedy and V. Pless, A coding-theoretic approach to extending designs, Discrete Math., 142 (1995), 155-168.  doi: 10.1016/0012-365X(94)00010-G.  Google Scholar  J.-L. Kim and V. Pless, Designs in additive codes over $GF(4)$, Des. Codes Cryptogr., 30 (2003), 187-199.  doi: 10.1023/A:1025484821641.  Google Scholar  R. Lidl and H. Niederreiter, Finite Fields, volume 20 of Encyclopedia of Mathematics and its Applications, Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, 1983. Google Scholar  J. Luo, Y. Tang and H. Wang, Cyclic codes and sequences: The generalized Kasami case, IEEE Trans. Inform. Theory, 56 (2010), 2130-2142.  doi: 10.1109/TIT.2010.2043783.  Google Scholar  F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-correcting Codes, I, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. Google Scholar  C. Reid and A. Rosa, Steiner systems ${S} (2, 4, v)$-a survey, The Electronic Journal of Combinatorics, DS18 (2010), 1–34. https://www.researchgate.net/publication/266996333. doi: 10.37236/39. Google Scholar  V. D. Tonchev, Codes and designs, In V. Pless and W. C. Huffman, editors, Handbook of Coding Theory, Vol. I, II, pages 1229–1267. North-Holland, Amsterdam, 1998. https://www.researchgate.net/publication/268549395. Google Scholar  V. D. Tonchev, Codes, In C. J. Colbourn and J. H. Dinitz, editors, Handbook of Combinatorial Designs, Discrete Mathematics and its Applications (Boca Raton), pages xxii+984. Chapman & Hall/CRC, Boca Raton, FL, second edition, 2007. Google Scholar
The weight distribution of ${\overline{{\mathcal{C}_1}^{\bot}}}^{\bot}$
 Weight Multiplicity $0$ $1$ $2^{2s-1}$ $29\times2^{6s-5}-33\times2^{4s-5}+17\times2^{2s-3}-2$ $2^{2s-1}-2^{s-1}$ $\frac{2}{15}\times2^{2s}(3\times2^{4s}+5\times2^{2s}-8)$ $2^{2s-1}+2^{s-1}$ $\frac{2}{15}\times2^{2s}(3\times2^{4s}+5\times2^{2s}-8)$ $2^{2s-1}-2^s$ $\frac{7}{3}\times2^{4s-4}(2^{2s}-1)$ $2^{2s-1}+2^s$ $\frac{7}{3}\times2^{4s-4}(2^{2s}-1)$ $2^{2s-1}-2^{s+1}$ $\frac{1}{15}\times2^{2s-4}(2^{4s-2}-5\times2^{2s-2}+1)$ $2^{2s-1}+2^{s+1}$ $\frac{1}{15}\times2^{2s-4}(2^{4s-2}-5\times2^{2s-2}+1)$ $2^{2s}$ $1$
 Weight Multiplicity $0$ $1$ $2^{2s-1}$ $29\times2^{6s-5}-33\times2^{4s-5}+17\times2^{2s-3}-2$ $2^{2s-1}-2^{s-1}$ $\frac{2}{15}\times2^{2s}(3\times2^{4s}+5\times2^{2s}-8)$ $2^{2s-1}+2^{s-1}$ $\frac{2}{15}\times2^{2s}(3\times2^{4s}+5\times2^{2s}-8)$ $2^{2s-1}-2^s$ $\frac{7}{3}\times2^{4s-4}(2^{2s}-1)$ $2^{2s-1}+2^s$ $\frac{7}{3}\times2^{4s-4}(2^{2s}-1)$ $2^{2s-1}-2^{s+1}$ $\frac{1}{15}\times2^{2s-4}(2^{4s-2}-5\times2^{2s-2}+1)$ $2^{2s-1}+2^{s+1}$ $\frac{1}{15}\times2^{2s-4}(2^{4s-2}-5\times2^{2s-2}+1)$ $2^{2s}$ $1$
The weight distribution of ${\overline{{\mathcal{C}_2}^{\bot}}}^{\bot}$ when $d' = d$
 Weight Multiplicity $0$ $1$ $2^{2s-1}-2^{s-1}$ $2^{2s}(2^s-1)(2^{2(s+d)}-2^{2s+d}-2^{2s}+2^{s+2d}-2^{s+d}+2^{2d})/(2^{2d}-1)$ $2^{2s-1}+2^{s-1}$ $2^{2s}(2^s-1)(2^{2(s+d)}-2^{2s+d}-2^{2s}+2^{s+2d}-2^{s+d}+2^{2d})/(2^{2d}-1)$ $2^{2s-1}-2^{s+d-1}$ $2^{2(s-d)}(2^{s+d}-1)(2^{2s}-1)/(2^{2d}-1)$ $2^{2s-1}+2^{s+d-1}$ $2^{2(s-d)}(2^{s+d}-1)(2^{2s}-1)/(2^{2d}-1)$ $2^{2s-1}$ $2(2^{3s-d}-2^{2(s-d)}+1)(2^{2s}-1)$ $2^{2s}$ $1$
 Weight Multiplicity $0$ $1$ $2^{2s-1}-2^{s-1}$ $2^{2s}(2^s-1)(2^{2(s+d)}-2^{2s+d}-2^{2s}+2^{s+2d}-2^{s+d}+2^{2d})/(2^{2d}-1)$ $2^{2s-1}+2^{s-1}$ $2^{2s}(2^s-1)(2^{2(s+d)}-2^{2s+d}-2^{2s}+2^{s+2d}-2^{s+d}+2^{2d})/(2^{2d}-1)$ $2^{2s-1}-2^{s+d-1}$ $2^{2(s-d)}(2^{s+d}-1)(2^{2s}-1)/(2^{2d}-1)$ $2^{2s-1}+2^{s+d-1}$ $2^{2(s-d)}(2^{s+d}-1)(2^{2s}-1)/(2^{2d}-1)$ $2^{2s-1}$ $2(2^{3s-d}-2^{2(s-d)}+1)(2^{2s}-1)$ $2^{2s}$ $1$
The weight distribution of ${\overline{{\mathcal{C}_2}^{\bot}}}^{\bot}$ when $d' = 2d$
 Weight Multiplicity $0$ $1$ $2^{2s-1}-2^{s-1}$ $2^{2s+3d}(2^s-1)(2^{2s}-2^{2(s-d)}-2^{2s-3d}+2^s-2^{s-d}+1)/(2^{2d}-1)(2^d+1)$ $2^{2s-1}+2^{s-1}$ $2^{2s+3d}(2^s-1)(2^{2s}-2^{2(s-d)}-2^{2s-3d}+2^s-2^{s-d}+1)/(2^{2d}-1)(2^d+1)$ $2^{2s-1}-2^{s+d-1}$ $2^{2s-d}(2^{2s}-1)(2^s+2^{s-d}+2^{s-2d}+1)/(2^d+1)^2$ $2^{2s-1}+2^{s+d-1}$ $2^{2s-d}(2^{2s}-1)(2^s+2^{s-d}+2^{s-2d}+1)/(2^d+1)^2$ $2^{2s-1}$ $2(2^{2s}-1)(2^{3s-d}-2^{3s-2d}+2^{3s-3d}-2^{3s-4d}+2^{3s-5d}+2^{2s-d}-2^{2s-2d+1} +2^{2s-3d}-2^{2s-4d}+1)$ $2^{2s-1}-2^{s+2d-1}$ $2^{2s-4d}(2^{s-d}-1)(2^{2s}-1)/(2^d+1)(2^{2d}-1)$ $2^{2s-1}+2^{s+2d-1}$ $2^{2s-4d}(2^{s-d}-1)(2^{2s}-1)/(2^d+1)(2^{2d}-1)$ $2^{2s}$ $1$
 Weight Multiplicity $0$ $1$ $2^{2s-1}-2^{s-1}$ $2^{2s+3d}(2^s-1)(2^{2s}-2^{2(s-d)}-2^{2s-3d}+2^s-2^{s-d}+1)/(2^{2d}-1)(2^d+1)$ $2^{2s-1}+2^{s-1}$ $2^{2s+3d}(2^s-1)(2^{2s}-2^{2(s-d)}-2^{2s-3d}+2^s-2^{s-d}+1)/(2^{2d}-1)(2^d+1)$ $2^{2s-1}-2^{s+d-1}$ $2^{2s-d}(2^{2s}-1)(2^s+2^{s-d}+2^{s-2d}+1)/(2^d+1)^2$ $2^{2s-1}+2^{s+d-1}$ $2^{2s-d}(2^{2s}-1)(2^s+2^{s-d}+2^{s-2d}+1)/(2^d+1)^2$ $2^{2s-1}$ $2(2^{2s}-1)(2^{3s-d}-2^{3s-2d}+2^{3s-3d}-2^{3s-4d}+2^{3s-5d}+2^{2s-d}-2^{2s-2d+1} +2^{2s-3d}-2^{2s-4d}+1)$ $2^{2s-1}-2^{s+2d-1}$ $2^{2s-4d}(2^{s-d}-1)(2^{2s}-1)/(2^d+1)(2^{2d}-1)$ $2^{2s-1}+2^{s+2d-1}$ $2^{2s-4d}(2^{s-d}-1)(2^{2s}-1)/(2^d+1)(2^{2d}-1)$ $2^{2s}$ $1$
The weight distribution of ${\mathcal{C}_2}$ when $d' = d$
 Weight Multiplicity $0$ $1$ $2^{2s-1}-2^{s-1}$ $2^{s-1}(2^{2s}-1)(2^{2(s+d)}-2^{2s+d}-2^{2s}+2^{s+2d}-2^{s+d}+2^{2d})/(2^{2d}-1)$ $2^{2s-1}+2^{s-1}$ $2^{s-1}(2^s-1)^2(2^{2(s+d)}-2^{2s+d}-2^{2s}+2^{s+2d}-2^{s+d}+2^{2d})/(2^{2d}-1)$ $2^{2s-1}-2^{s+d-1}$ $2^{s-d-1}(2^{s+d}-1)(2^{2s}-1)(2^{s-d}+1)/(2^{2d}-1)$ $2^{2s-1}+2^{s+d-1}$ $2^{s-d-1}(2^{s+d}-1)(2^{2s}-1)(2^{s-d}-1)/(2^{2d}-1)$ $2^{2s-1}$ $(2^{3s-d}-2^{2(s-d)}+1)(2^{2s}-1)$
 Weight Multiplicity $0$ $1$ $2^{2s-1}-2^{s-1}$ $2^{s-1}(2^{2s}-1)(2^{2(s+d)}-2^{2s+d}-2^{2s}+2^{s+2d}-2^{s+d}+2^{2d})/(2^{2d}-1)$ $2^{2s-1}+2^{s-1}$ $2^{s-1}(2^s-1)^2(2^{2(s+d)}-2^{2s+d}-2^{2s}+2^{s+2d}-2^{s+d}+2^{2d})/(2^{2d}-1)$ $2^{2s-1}-2^{s+d-1}$ $2^{s-d-1}(2^{s+d}-1)(2^{2s}-1)(2^{s-d}+1)/(2^{2d}-1)$ $2^{2s-1}+2^{s+d-1}$ $2^{s-d-1}(2^{s+d}-1)(2^{2s}-1)(2^{s-d}-1)/(2^{2d}-1)$ $2^{2s-1}$ $(2^{3s-d}-2^{2(s-d)}+1)(2^{2s}-1)$
The weight distribution of ${\mathcal{C}_2}$ when $d' = 2d$
 Weight Multiplicity $0$ $1$ $2^{2s-1}-2^{s-1}$ $\frac{2^{s+3d-1}(2^{2s}-1)(2^{2s}-2^{2(s-d)}-2^{2s-3d}+2^s-2^{s-d}+1)}{(2^{2d}-1)(2^d+1)}$ $2^{2s-1}+2^{s-1}$ $\frac{2^{2s+3d-1}(2^s-1)^2(2^{2s}-2^{2(s-d)}-2^{2s-3d}+2^s-2^{s-d}+1)}{(2^{2d}-1)(2^d+1)}$ $2^{2s-1}-2^{s+d-1}$ $2^{s-1}(2^{2s}-1)(2^s+2^{s-d}+2^{s-2d}+1)(2^{s-d}+1)/(2^d+1)^2$ $2^{2s-1}+2^{s+d-1}$ $2^{s-1}(2^{2s}-1)(2^s+2^{s-d}+2^{s-2d}+1)(2^{s-d}-1)/(2^d+1)^2$ $2^{2s-1}$ $(2^{2s}-1)(2^{3s-d}-2^{3s-2d}+2^{3s-3d}-2^{3s-4d}+2^{3s-5d} +2^{2s-d}-2^{2s-2d+1}+2^{2s-3d}-2^{2s-4d}+1)$ $2^{2s-1}-2^{s+2d-1}$ $2^{s-2d-1}(2^{s-d}-1)(2^{2s}-1)(2^{s-2d}+1)/(2^d+1)(2^{2d}-1)$ $2^{2s-1}+2^{s+2d-1}$ $2^{s-2d-1}(2^{s-d}-1)(2^{2s}-1)(2^{s-2d}-1)/(2^d+1)(2^{2d}-1)$
 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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