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Cryptography and Communications
Published in: Cryptography and Communications 6/2021

17-02-2021

The linear codes of t-designs held in the Reed-Muller and Simplex codes

Authors: Cunsheng Ding, Chunming Tang

Published in: Cryptography and Communications | Issue 6/2021

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Abstract

A fascinating topic of combinatorics is the study of t-designs, which has a very long history. The incidence matrix of a t-design generates a linear code over GF(q) for any prime power q, which is called the linear code of the t-design over GF(q). On the other hand, some linear codes hold t-designs with t ≥ 1. The purpose of this paper is to study the linear codes of t-designs held in the Reed-Muller and Simplex codes. Some general theory for the linear codes of t-designs held in linear codes is presented. Several open problems are also presented.
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Literature
1.
Assmus Jr., E.F., Key, J.D.: Designs and Their Codes. Cambridge University Press, Cambridge (1992)CrossRef
2.
Assmus Jr., E.F., Key, J.D.: Polynomial Codes and Finite Geometries. In: Pless, V.S., Huffman, W.C. (eds.) Handbook of Coding Theory, Vol. II, pp. 1269-1343. Elsevier, Amsterdam (1998)
3.
4.
Beth, T., Jungnickel, D., Lenz, H.: Design Theory. Cambridge University Press, Cambridge (1999)CrossRef
5.
Ceccherini, P.V., Hirschfeld, J.W.P.: The dimension of projective geometry codes. Discrete Math. 107, 117–126 (1992)MathSciNetCrossRef
6.
Charpin, P.: Codes cycliques étendus affines-invariants et antichaînes d’un ensemble partiellement ordonné. Discrete Math. 80, 229–247 (1990)MathSciNetCrossRef
7.
8.
Ding, C.: Codes from Difference Sets. World Scientific, Singapore (2015)
9.
10.
Ding, C., Munemasa, A., Tonchev, V.D.: Bent vectorial functions, codes and designs. IEEE Trans. Inf. Theory 65(11), 7533–7541 (2019)MathSciNetCrossRef
11.
Ding, C., Tang, C., Tonchev, V.D.: Linear codes of 2-designs associated with subcodes of the ternary generalized Reed-Muller codes. Des. Codes Cryptogr. 88(4), 625–641 (2020)MathSciNetCrossRef
12.
Hamada, N.: The rank of the incidence matrix of points and d-flats in finite geometries. J. Sci. Hiroshima Univ. Ser. A-I 32, 381–396 (1968)MathSciNetMATH
13.
Hamada, N.: On the p-rank of the incidence matrix of a balanced or partially balanced incomplete block design and its applications to error correcting codes. Hiroshima Math. J. 3, 153–226 (1973)MathSciNetCrossRef
14.
Huffman, W.C., Pless, V.: Fundamentals of Error-Correcting Codes. Cambridge University Press, Cambridge (2003)CrossRef
15.
Jungnickel, D., Magliveras, S.S., Tonchev, V.D., Wassermann, A., et al.: On Classifying Steiner Triple Systems by Their 3-Rank. In: Blömer, J. (ed.) MACIS 2017, LNCS 10693, pp. 295–305. Springer, Cham (2017)
16.
17.
Li, S.: On the weight distribution of second order Reed-Muller codes and their relatives. Des. Codes Cryptogr. 80, 2447–2460 (2019)MathSciNetCrossRef
18.
McGuire, G., Ward, H.N.: Characterization of certain minimal rank designs. J. Comb. Theory Ser. A 83, 42–56 (1998)MathSciNetCrossRef
19.
Serre, J.P.: Letter to M. Tsafasman, dated July 29 (1989)
20.
Tonchev, V.D.: Quasi-symmetric designs, codes, quadrics, and hyperplane sections. Geom. Dedicata. 48, 295–308 (1993)MathSciNetCrossRef
21.
Tonchev, V.D.: Codes and Designs. In: Pless, V.S., Huffman, W.C. (eds.) Handbook of Coding Theory, Vol. II, pp. 1229-1268. Elsevier, Amsterdam (1998)
22.
Tonchev, V.D.: Linear perfect codes and a characterization of the classical designs. Des. Codes Cryptogr. 17, 121–128 (1999)MathSciNetCrossRef
23.
Tonchev, V.D. Colbourn, C.J., Dinitz, J.H. (eds.) . CRC Press, New York (2007)
Metadata
Title
The linear codes of t-designs held in the Reed-Muller and Simplex codes
Authors
Cunsheng Ding
Chunming Tang
Publication date
17-02-2021
Publisher
Springer US
Published in
Cryptography and Communications / Issue 6/2021
Print ISSN: 1936-2447
Electronic ISSN: 1936-2455
DOI
https://doi.org/10.1007/s12095-021-00470-6

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