Abstract
MDS constacyclic codes over finite fields are important in both theory and practice. In this paper, all [q + 1,2,q] and [q + 1,3,q − 1] MDS λ-constacyclic codes over GF(q) are characterized, three classes of [q + 1,4,q − 2] MDS λ-constacyclic codes over GF(q) are constructed, and four classes of [q + 1,k,q − k + 2] MDS λ-constacyclic codes over GF(q) for variable k are presented. Rationales for distinguishing the class of cyclic codes, the class of λ-constacyclic codes with λ≠ 1, and the class of non-constacyclic linear codes are given. Some applications of MDS codes in cryptography are also summarised.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Blackford, T.: Negacyclic duadic codes. Finite Fields Appl. 14, 930–943 (2008)
Casse, L. R. A., Glynn, D. G.: The solution to Beniamino Segre’s problem ir,q, r = 3, q = 2h. Geom. Ded. 13, 157–163 (1982)
Chen, B., Dinh, H. Q., Fan, Y., Ling, S.: Polyadic constacyclic codes. IEEE Trans. Inf. Theory 61(9), 4895–4904 (2015)
Chen, B., Fan, Y., Lin, L., Liu, H.: Constacyclic codes over finite fields. Finite Fields Appl. 18, 1217–1231 (2012)
Dahl, C., Pedersen, J. P.: Cyclic and pseudo-cyclic MDS codes of length q + 1. J. Comb. Theory Ser. A 59, 130–133 (1992)
Danev, D., Dodunekov, S., Radkova, D.: A family of constacyclic ternary quasi-perfect codes with covering radius 3. Des. Codes Cryptogr. 59, 111–118 (2011)
Da Rocha, V.: Maximum distance separable multilevel codes. IEEE Trans. Inf. Theory 30(3), 547–548 (1984)
Ding, C., Helleseth, T., Klø ve, T., Wang X.: A general construction of authentication codes. IEEE Trans. Inf. Theory 53(6), 2229–2235 (2007)
Ding, C., Laihonen, T., Renvall, A.: Linear multi-secret sharing schemes and error-correcting codes. J. Universal Computer Science 3(9), 1023–1036 (1997)
Ding, C., Pei, D., Salomaa, A.: Chinese Remainder Theorem: Applications in Computing, Coding and Cryptography. World Scientific, Singapore (1996)
Dinh, H. Q.: Constacyclic codes of length ps over \(\mathbb {F}_{p^{m}}+u \mathbb {F}_{p^{m}}\). J. Algebra 324, 940–950 (2010)
Georgiades, J.: Cyclic (q + 1,k)-codes of odd order q and even dimension k are not optimal. Atti. Sent. Mat. Fis. Univ. Modena 30, 284–285 (1982)
Gupta, K. C., Pandey, S. K., Ray, I. G., Samanta, S.: Cryptographically significant MDS matrices over finite feilds: a brief survey and some generalized results. Adv. Math. Commun. 13(4), 779–843 (2019)
Krishna, A., Sarwate, D. V.: Pseudocyclic maximum-distance-separable codes. IEEE Trans. Inf. Theory 36(4), 880–884 (1990)
Koroglu, M. E., San, M.: On MDS negacyclic LCD codes. Filomat 33(1), 1–12 (2019)
La Guardia, G. G.: On optimal constacyclic codes. Linear Algebra Appl. 496, 594–610 (2016)
Li, S., Xiong, M., Ge, G.: Pseudocyclic codes and the construction of quantum MDS codes. IEEE Trans. Inf. Theory 62(4), 1703–1710 (2016)
Liu, Y., Li, R., Lv, L., Ma, Y.: A class of constacyclic BCH codes and new quantum codes. Quantum Inf. Process. 16(66), 1–16 (2017)
MacWilliams, F. J., Sloane, N. J. A.: The Theory of Error-Correcting Codes. North-Holland, Amsterdam (1977)
Maruta, T.: On the uniqueness of cyclic MDS codes, Atti Sem. Mat. Fis. Univ. Modena XL 83–88 (1992)
Maruta, T.: On the existence of cyclic and pseudo-cyclic MDS codes. Europ. J. Combinatorics 19, 159–174 (1998)
Mi, J., Cao, X.: Constructing MDS Galois self-dual constacyclic codes over finite fields. Discrete Math. 344(6), 1–15 (2021)
Pedersen, J. P., Dahl, C.: Classification of pseudo-cyclic MDS codes. IEEE Trans. Inf. Theory 37(2), 365–370 (1991)
Peterson, W. W., Weldon, E. J.: Error-Correcting Codes, 2nd edn. MIT Press, Cambridge (1972)
Renvall, A., Ding, C.: The access structure of some secret sharing schemes. Information Security and Privacy, LNCS 1172, pp 67–78. Springer, Berlin (1996)
Renvall, A., Ding, C.: A nonlinear secret sharing scheme. Information Security and Privacy. LNCS 1172, pp 56–65. Springer, Berlin (1996)
Sun, Z., Zhu, S., Wang, L.: A class of constacyclic BCH codes. Cryptogr. Commun. 12, 265–284 (2020)
Tang, C., Wang, Q., Ding, C.: The subfield subcodes and subfield codes of a family of MDS codes. IEEE Trans. Inf. Theory 68(9), 5792–5801 (2022)
Acknowledgements
The authors sincerely thank the editors and the reviewers for their helpful comments and valuable suggestions, which have improved the presentation of this paper. The second author would like to thank Tania Sidana for helpful discussions on some MDS cyclic codes.
Funding
X. Wang’s research was supported by the National Natural Science Foundation of China under Grant Number 12001175. C. Ding’s research was supported by the Hong Kong Research Grants Council, Proj. No. 16301522. H. Liu’s research was supported by the National Natural Science Foundation of China under Grant Number 11871025. D. Zheng’s research was supported by the National Natural Science Foundation of China under Grant Number 11971156.
Author information
Authors and Affiliations
Contributions
These authors contributed equally to this work.
Corresponding author
Ethics declarations
Conflict of Interests
The authors have no conficts of interest to declare that are relevant to the content of this article.
Additional information
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Wang, X., Ding, C., Liu, H. et al. MDS constacyclic codes of length q + 1 over GF(q). Cryptogr. Commun. 16, 21–48 (2024). https://doi.org/10.1007/s12095-022-00624-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12095-022-00624-0