Abstract
Coding theory and combinatorial t-designs have close connections and interesting interplay. One of the major approaches to the construction of combinatorial t-designs is the employment of error-correcting codes. As we all known, some t-designs have been constructed with this approach by using certain linear codes in recent years. However, only a small number of infinite families of cyclic codes holding an infinite family of 3-designs are reported in the literature. The objective of this paper is to study an infinite family of antiprimitive cyclic codes and determine their parameters. By the parameters of these codes and their dual, some infinite family of 3-designs are presented and their parameters are also explicitly determined. In particular, the complements of the supports of the minimum weight codewords in the studied cyclic code form a Steiner system. Furthermore, we show that the infinite family of cyclic codes admit 3-transitive automorphism groups.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Chien R.T.: Cyclic decoding procedure for the Bose-Chaudhuri-Hocquenghem codes. IEEE Trans. Inf. Theory 10(4), 357–363 (1964).
Ding C.: An infinite family of Steiner systems from cyclic codes. J. Comb. Des. 26, 127–144 (2018).
Ding C.: A sequence construction of cyclic codes over finite fields. Cryptogr. Commun. 10(2), 319–341 (2018).
Ding C.: Infinite families of 3-designs from a type of five-weight code. Des. Codes Cryptogr. 66(3), 703–719 (2018).
Ding C., Helleseth T.: Optimal ternary cyclic codes from monomials. IEEE Trans. Inf. Theory 59(9), 5898–5904 (2013).
Ding C., Li C.: Infinite families of \(2\)-designs and \(3\)-designs from linear codes. Discret. Math. 340(10), 2415–2431 (2017).
Ding C., Tang C.: Infinite families of near MDS codes holding t-designs. IEEE Trans. Inf. Theory 66(9), 5419–5428 (2020).
Ding C., Tang C.: The linear codes of t-designs held in the Reed-Muller and Simplex codes. Cryptogr. Commun. 13(6), 927–949 (2021).
Ding C., Tang C.: Designs from Linear Codes, 2nd edn World Scientific, Singapore (2022).
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).
Du X., Wang R., Fan C.: Infinite families of 2-designs from a class of cyclic codes with two non-zeros (2019). arXiv:1904.04242.
Du Z., Li C., Mesnager S.: Constructions of self-orthogonal codes from hulls of BCH codes and their parameters. IEEE Trans. Inf. Theory 66(11), 6774–6785 (2020).
Forney G.D.: On decoding BCH codes. IEEE Trans. Inf. Theory 11(4), 549–557 (1995).
Huffman W.C., Pless V.: Fundamentals of Error-Correcting Codes. Cambridge University Press, Cambridge (2003).
Liu H., Ding C.: Infinite families of 2-designs from GA1(q) actions (2017). ArXiv:1707.02003.
Liu Q., Ding C., Mesnager S., et al.: On infinite families of narrow-sense antiprimitive BCH codes admitting 3-transitive automorphism groups and their consequences. IEEE Trans. Inf. Theory (2022). https://doi.org/10.1109/TIT.2021.3139687.
Prange E.: Some cyclic error-correcting codes with simple decoding algorithms. In: Air Force Cambridge Research Center-TN-58-156, Cambridge, MA, April (1958).
Tang C., Ding C.: An infinite family of linear codes supporting \(4\)-designs. IEEE Trans. Inf. Theory 67(1), 244–254 (2021).
Tang C., Ding C., Xiong M.: Steiner systems \(S(2, 4, \frac{3^m-1}{2})\) and 2-designs from ternary linear codes of length \(\frac{3^m-1}{2}\). Des. Codes Cryptogr. 87(12), 2793–2811 (2019).
Tang C., Ding C., Xiong M.: Codes, differentially \(\delta \)-uniform functions, and \(t\)-designs. IEEE Trans. Inf. Theory 66(6), 3691–3703 (2020).
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).
Tonchev V.D.: Codes. In: Colbourn C.J., Dinitz J.H. (eds.) Handbook of Combinatorial Designs, 2nd edn, pp. 677–701. CRC Press, New York (2007).
Wang R., Du X., Fan C.: Infinite families of 2-designs from a class of non-binary Kasami cyclic codes (2019). arXiv:1912.04745.
Yan H.: A class of primitive BCH codes and their weight distribution. Appl. Algebra Eng. Commun. Comput. 29(1), 1–11 (2018).
Yan H., Zhou Z., Du X.: A family of optimal ternary cyclic codes from the Niho-type exponent. Finite Fields Their Appl. 54, 101–112 (2018).
Zhou Z., Ding C.: Seven classes of three-weight cyclic codes. IEEE Trans. Commun. 61(10), 4120–4126 (2013).
Acknowledgements
The authors are very grateful to the reviewers and the Editor, for their comments and suggestions that improved the presentation and quality of this paper. This paper was supported by the National Natural Science Foundation of China under Grant Numbers 12171162 and 11871058 and the Basic Research Project of Science and Technology Plan of Guangzhou city of China under Grant Number 202102020888.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by V. D. Tonchev.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Xiang, C., Tang, C. & Liu, Q. An infinite family of antiprimitive cyclic codes supporting Steiner systems \(S(3,8, 7^m+1)\). Des. Codes Cryptogr. 90, 1319–1333 (2022). https://doi.org/10.1007/s10623-022-01032-4
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10623-022-01032-4