main-content

## Swipe to navigate through the articles of this issue

Published in:

24-06-2020 | Original Paper

# Infinite families of 2-designs from linear codes

Authors: Xiaoni Du, Rong Wang, Chunming Tang, Qi Wang

## Abstract

Interplay between coding theory and combinatorial t-designs has attracted a lot of attention. It is well known that the supports of all codewords of a fixed Hamming weight in a linear code may hold a t-design. In this paper, we first settle the weight distributions of two classes of linear codes, and then determine the parameters of infinite families of 2-designs held in these codes.
Appendix
Available only for authorised users
Literature
1.
Assmus Jr., E.F., Key, J.D.: Designs and Their Codes, Cambridge Tracts in Mathematics, vol. 103. Cambridge University Press, Cambridge (1992) CrossRef
2.
Beth, T., Jungnickel, D., Lenz, H.: Design Theory, Vol. II, Encyclopedia of Mathematics and Its Applications, vol. 78, 2nd edn. Cambridge University Press, Cambridge (1999) MATH
3.
Colbourn, C.J., Mathon, R.: Steiner systems. In: Colbourn, C.J. and Dinitz. J.H. (eds.) Handbook of Combinatorial Designs, 2nd edn, pp. 128–135. Chapman and Hall, Boca Raton (2006)
4.
Coulter, R.S.: Further evaluations of Weil sums. Acta Arith. 86(3), 217–226 (1998)
5.
Ding, C.: Codes from Difference Sets. World Scientific, Singapore (2015)
6.
Ding, C.: Designs from Linear Codes. World Scientific, Singapore (2018) CrossRef
7.
Ding, C.: Infinite families of 3-designs from a type of five-weight code. Des. Codes Cryptogr. 86(3), 703–719 (2018)
8.
Ding, C., Li, C.: Infinite families of 2-designs and 3-designs from linear codes. Discrete Math. 340(10), 2415–2431 (2017)
9.
Ding, K., Ding, C.: A class of two-weight and three-weight codes and their applications in secret sharing. IEEE Trans. Inf. Theory 61(11), 5835–5842 (2015)
10.
Du, X., Wang, R., Fan, C.: Infinite families of 2-designs from a class of cyclic codes with two non-zeros. ArXiv preprint arXiv:​1904.​04242 (2019)
11.
Du, X., Wang, R., Tang, C., Wang, Q.: Infinite families of 2-designs from two classes of binary cyclic codes with three non-zeros. ArXiv preprint arXiv:​1903.​08153 (2019)
12.
Feng, K., Luo, J.: Weight distribution of some reducible cyclic codes. Finite Fields Appl. 14(2), 390–409 (2008)
13.
Fitzgerald, R.W., Yucas, J.L.: Sums of Gauss sums and weights of irreducible codes. Finite Fields Appl. 11(1), 89–110 (2005)
14.
Huffman, W.C., Pless, V.: Fundamentals of Error-Correcting Codes. Cambridge University Press, Cambridge (2003) CrossRef
15.
Ireland, K., Rosen, M.: A Classical Introduction to Modern Number Theory, Graduate Texts in Mathematics, vol. 84, 2nd edn. Springer, New York (1990) CrossRef
16.
Kasami, T., Lin, S., Peterson, W.W.: Some results on cyclic codes which are invariant under the affine group and their applications. Inf. Control 11, 475–496 (1967)
17.
Kennedy, G.T., Pless, V.: A coding-theoretic approach to extending designs. Discrete Math. 142(1–3), 155–168 (1995)
18.
Kim, J.L., Pless, V.: Designs in additive codes over $$GF(4)$$. Des. Codes Cryptogr. 30(2), 187–199 (2003)
19.
Lidl, R., Niederreiter, H.: Finite Fields, Encyclopedia of Mathematics and Its Applications, vol. 20, 2nd edn. Cambridge University Press, Cambridge (1997)
20.
MacWilliams, F.J., Sloane, N.J.A.: The Theory of Error-Correcting Codes: I. North-Holland Mathematical Library, vol. 16. North-Holland Publishing Co., Amsterdam (1977)
21.
Reid, C., Rosa, A.: Steiner systems $$S (2, 4, v)$$-a survey. Electron. J. Comb. #DS18, 1–34 (2010)
22.
Tonchev, V.D.: Codes and designs. In: Pless, V., Huffman, W.C. (eds.) Handbook of Coding Theory, vol. I and II, pp. 1229–1267. North-Holland, Amsterdam (1998)
23.
Tonchev, V.D.: Codes. In: Colbourn, C.J., Dinitz, J.H. (eds.) Handbook of Combinatorial Designs, Discrete Mathematics and Its Applications (Boca Raton), 2nd edn. Chapman & Hall, Boca Raton (2007)
24.
van der Vlugt, M.: Hasse–Davenport curves, Gauss sums, and weight distributions of irreducible cyclic codes. J. Number Theory 55(2), 145–159 (1995)
25.
Xiang, C., Ling, X., Wang, Q.: Combinatorial $$t$$-designs from quadratic functions. Des. Codes Cryptogr. 88(3), 553–565 (2020)
Title
Infinite families of 2-designs from linear codes
Authors
Xiaoni Du
Rong Wang
Chunming Tang
Qi Wang
Publication date
24-06-2020
Publisher
Springer Berlin Heidelberg
Published in
Applicable Algebra in Engineering, Communication and Computing / Issue 3/2022
Print ISSN: 0938-1279
Electronic ISSN: 1432-0622
DOI
https://doi.org/10.1007/s00200-020-00438-8

Go to the issue