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Designs, Codes and Cryptography

Erschienen in:

14.11.2023

Mutually disjoint Steiner systems from BCH codes

verfasst von: Qianqian Yan, Junling Zhou

Erschienen in: Designs, Codes and Cryptography | Ausgabe 4/2024

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Abstract

Liu et al. (IEEE Trans Inf Theory 68:3096–3107, 2022) investigated a class of BCH codes \(\mathcal {C}_{(q,q+1,\delta ,1)}\) with \(q=\delta ^m\) a prime power and proved that the set \(\mathcal {B}_{\delta +1}\) of supports of the minimum weight codewords supports a Steiner system \({{\text {S}}}(3,\delta +1,q+1)\). In this paper, we give an equivalent formulation of \(\mathcal {B}_{\delta +1}\) in terms of elementary symmetric polynomials and then construct a number of mutually disjoint Steiner systems S\((3,\delta +1,\delta ^m+1)\) when m is even and a number of mutually disjoint G-designs G\(\big ({\frac{\delta ^m+1}{\delta +1}},\delta +1,\delta +1,3\big )\) when m is odd. In particular, the existence of three mutually disjoint Steiner systems \({{\text {S}}}(3,5,4^m+1)\) or three mutually disjoint G-designs G\(\big ({\frac{4^m+1}{5}},5,5,3\big )\) is established.

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Angata M., Shiromoto K.: Mutually disjoint \(5\)-designs from Pless symmetry codes. J. Stat. Theory Pract. 6, 78–87 (2012).MathSciNetCrossRef

Araya M., Harada M.: Mutually disjoint Steiner systems \(\text{S}(5, 8, 24)\) and \(5\)-\((24,12,48)\) designs. Electron. J. Comb. 17, N1 (2010).

Araya M., Harada M., Tonchev V.D., Wassermann A.: Mutually disjoint designs and new \(5\)-designs derived from groups and codes. J. Comb. Des. 18, 305–317 (2010).MathSciNetCrossRef

Ding C., Tang C.: Infinite families of near MDS codes holding \(t\)-designs. IEEE Trans. Inf. Theory 66, 5419–5428 (2020).MathSciNetCrossRef

Ding C., Tang C., Tonchev V.D.: The projective general linear group PGL\((2,2^m)\) and linear codes of length \(2^m+1\). Des. Codes Cryptogr. 89, 1713–1734 (2021).MathSciNetCrossRef

Etzion T.: Partitions for quadruples. Ars Comb. 36, 296–308 (1993).MathSciNet

Etzion T., Hartman A.: Towards a large set of Steiner quadruple systems. SIAM J. Discret. Math. 4, 182–195 (1991).MathSciNetCrossRef

Fang J., Chang Y.: Mutually disjoint \(t\)-designs and \(t\)-SEEDs from extremal doubly-even self-dual codes. Des. Codes Cryptogr. 73, 769–780 (2014).MathSciNetCrossRef

Fang J., Chang Y.: Mutually disjoint \(5\)-designs and \(5\)-spontaneous emission error designs from extremal ternary self-dual codes. J. Comb. Des. 23, 78–89 (2015).MathSciNetCrossRef

10.

Heinemann E.R.: Generalized Vandermonde determinants. Trans. Am. Math. Soc. 31, 464–476 (1929).MathSciNetCrossRef

11.

Huffman W.C., Pless V.: Fundamentals of Error-Correcting Codes. Cambridge University Press, Cambridge (2003).CrossRef

12.

Jimbo M., Shiromoto K.: A construction of mutually disjoint Steiner systems from isomorphic Golay codes. J. Comb. Theory Ser. A 116, 1245–1251 (2009).MathSciNetCrossRef

13.

Kramer E.S., Magliveras S.S.: Some mutually disjoint Steiner systems. J. Comb. Theory Ser. A 17, 39–43 (1974).MathSciNetCrossRef

14.

Lin S., Costello D.J.: Error Control Coding: Fundamentals and Applications. Prentice Hall, Upper Saddle River (2004).

15.

Liu Q., Ding C., Mesnager S., Tang C., Tonchev V.D.: On infinite families of narrow-sense antiprimitive BCH codes admitting \(3\)-transitive automorphism groups and their consequences. IEEE Trans. Inf. Theory 68, 3096–3107 (2022).MathSciNetCrossRef

16.

Lu J.: On large sets of disjoint Steiner triple systems, I–III. J. Comb. Theory Ser. A 34, 140–182 (1983).MathSciNetCrossRef

17.

Lu J.: On large sets of disjoint Steiner triple systems, IV–VI. J. Comb. Theory Ser. A 37, 136–192 (1984).MathSciNetCrossRef

18.

Mendes A., Remmel J.: Counting with Symmetric Functions. Springer, New York (2015).CrossRef

19.

Muir T.: A Treatise on the Theory of Determinants. Dover Publications Inc, New York (1960).

20.

Tang C., Ding C.: An infinite family of linear codes supporting \(4\)-designs. IEEE Trans. Inf. Theory 67, 244–254 (2021).MathSciNetCrossRef

21.

Teirlinck L.: Locally trivial \(t\)-designs and \(t\)-designs without repeated blocks. Discret. Math. 77, 345–356 (1989).MathSciNetCrossRef

22.

Van de Vel H.: Numerical treatment of a generalized Vandermonde system of equations. Linear Algebra Appl. 17, 149–179 (1977).MathSciNetCrossRef

Titel: Mutually disjoint Steiner systems from BCH codes
verfasst von: Qianqian Yan
Junling Zhou
Publikationsdatum: 14.11.2023
Verlag: Springer US
Erschienen in: Designs, Codes and Cryptography / Ausgabe 4/2024
Print ISSN: 0925-1022
Elektronische ISSN: 1573-7586
DOI: https://doi.org/10.1007/s10623-023-01319-0

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Mutually disjoint Steiner systems from BCH codes

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