February  2022, 16(1): 157-168. 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  February 2022 Early access  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, 2022, 16 (1) : 157-168. doi: 10.3934/amc.2020106
References:
[1] E. F. AssmusJr. and J. D. Key, Designs and Their Codes, volume 103 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1992.  doi: 10.1017/CBO9781316529836.
[2]

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.

[3] T. BethD. Jungnickel and H. Lenz, Design Theory, Cambridge University Press, Cambridge, 1986. 
[4]

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.

[5]

C. Ding, Codes from Difference Sets, World Scientific, 2015.

[6]

C. Ding, Designs from Linear Codes, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2019. doi: 10.1142/11101.

[7]

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.

[8]

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.

[9] W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511807077.
[10]

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.

[11]

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.

[12]

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.

[13]

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.

[14]

J. LuoY. 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.

[15]

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-correcting Codes, I, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.

[16]

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.

[17]

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.

[18]

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.

show all references

References:
[1] E. F. AssmusJr. and J. D. Key, Designs and Their Codes, volume 103 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1992.  doi: 10.1017/CBO9781316529836.
[2]

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.

[3] T. BethD. Jungnickel and H. Lenz, Design Theory, Cambridge University Press, Cambridge, 1986. 
[4]

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.

[5]

C. Ding, Codes from Difference Sets, World Scientific, 2015.

[6]

C. Ding, Designs from Linear Codes, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2019. doi: 10.1142/11101.

[7]

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.

[8]

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.

[9] W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511807077.
[10]

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.

[11]

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.

[12]

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.

[13]

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.

[14]

J. LuoY. 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.

[15]

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-correcting Codes, I, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.

[16]

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.

[17]

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.

[18]

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.

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 $
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 $
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 $
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) $
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) $
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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