Abstract
Negacyclic BCH codes are an important subclass of negacyclic codes and have good parameters. Inspired by the recent work on cyclic codes published in Wu et al. (Finite Fields Appl 60:101581, 2019), the objective of this paper is to investigate the parameters of the narrow-sense negacyclic BCH codes of length \(n=\frac{q^{2m}-1}{q+1}\) over \({\textrm{GF}}(q)\), where q is an odd prime power. For \(2\le \delta \le \left\lfloor \frac{q^{m+1}-q}{q+1} \right\rfloor +2\), the dimension of the narrow-sense negacyclic BCH codes of length n with designed distance \(\delta \) is determined. For \(2\le \delta \le \frac{q^{2m-1}+1}{q+1}\), a lower bound on the minimum distance of the dual codes of the narrow-sense negacyclic BCH codes of length n with designed distance \(\delta \) is presented. Compared with cyclic codes, we obtain some negacyclic BCH codes with better parameters.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data availability
Not applicalble.
Code availability
Not applicalble.
References
Augot D., Levy-dit-Vehel F.: Bounds on the minimum distance of the duals of BCH codes. IEEE Trans. Inf. Theory 42(4), 1257–1260 (1996).
Berlekamp E.R.: Negacyclic codes for the Lee metric. In: Proceedings of the Conference Combinatorial Mathematics and Its Applications, Chapel Hill, NC, pp. 298–316 (1968).
Blackford T.: Negacyclic duadic codes. Finite Fields Appl. 14, 930–943 (2008).
Danev D., Dodunekov S., Radkova D.: A family of constacyclic ternary quasi-perfect codes with covering radius. Des. Codes Cryptogr. 59, 111–118 (2011).
Dumer I.I., Zinoviev V.A.: Some new maximal codes over Galois field GF(4). Probl. Inf. Trans. 14(3), 24–34 (1978).
Fan M., Li C., Ding C.: The Hermitian dual codes of several classes of BCH codes. IEEE Trans. Inf. Theory 69(7), 4484–4497 (2023).
Gong B., Ding C., Li C.: The dual codes of several classes of BCH codes. IEEE Trans. Inf. Theory 68(2), 953–964 (2022).
Goppa V.D.: A new class of linear error-correcting codes. Probl. Peredachi Inf. 6, 24–30 (1970).
Grassl M.: Bounds on the minimum distance of linear codes and quantum codes. http://www.codetables.de
Guo G., Li R., Liu Y., Wang J.: A family of negacyclic BCH codes of length \(n=\frac{q^{2m}-1}{2}\). Cryptogr. Commun. 12, 187–203 (2020).
Kai X., Li P., Zhu S.: Construction of quantum negacyclic BCH codes. Int. J. Quantum Inf. 16, 1850059 (2018).
Krasikov I., Litsyn S.: On the distance distribution of duals of BCH codes. IEEE Trans. Inf. Theory 45(1), 247–250 (1999).
Krasikov I., Litsyn S.: On the distance distributions of BCH codes and their duals. Des. Codes Cryptogr. 23(2), 223–232 (2001).
Krishna A., Sarwate D.V.: Pseudocyclic maximum-distance-separable codes. IEEE Trans. Inf. Theory 36(4), 880–884 (1990).
Liu H., Ding C., Li C.: Dimensions of three types of BCH codes over GF(q). Discret. Math. 340, 1910–1927 (2017).
MacWilliams F.J., Sloane N.J.A.: The Theory Error-Correcting Codes. North-Holland, Amsterdam (1977).
Sun Z., Ding C.: Several families of ternary negacyclic codes and their duals (2023). arXiv:2301.09783v2
Wang L., Sun Z., Zhu S.: Hermitian dual-containing narrow-sense constacyclic BCH codes and quantum codes. Quantum Inf. Process. 18, 323 (2019).
Wang J., Li R., Liu Y., Guo G.: Some negacyclic BCH codes and quantum codes. Quantum Inf. Process. 19, 74 (2020).
Wang X., Wang J., Li C., Wu Y.: Two classes of narrow-sense BCH codes and their duals. IEEE Trans. Inf. Theory (2023). https://doi.org/10.1109/TIT.2023.3310193.
Wang X., Sun Z., Ding C.: Two families of negacyclic BCH codes. Des. Codes Cryptogr. 91, 2395–2420 (2023).
Wu P., Li C., Peng W.: On some cyclic codes of length \(\frac{q^{2m}-1}{q+1}\). Finite Fields Appl. 60, 101581 (2019).
Zhou Y., Kai X., Zhu S., Li J.: On the minimum distance of negacyclic codes with two zeros. Finite Fields Appl. 55, 134–150 (2019).
Zhu S., Sun Z., Li P.: A class of negacyclic BCH codes and its application to quantum codes. Des. Codes Cryptogr. 86(10), 2139–2165 (2018).
Zinoviev V.A., Litsyn S.N.: On the dual distance of BCH codes. Probl. Peredachi Inf. 22(4), 29–34 (1986).
Acknowledgements
The authors would like to thank the two anonymous reviewers and the Editor Prof. V. D. Tonchev for their valuable comments which help to improve the presentation of this manuscript. All the code examples in this paper were computed with the Magma software package.
Funding
Z. Sun’s research was supported by the National Natural Science Foundation of China under Grant Number Nos. 62002093 and 61972126. S. Zhu’s research was supported by the National Natural Science Foundation of China under Grant Number Nos. 12171134 and U21A20428. The work was supported by Key research project of Anhui Provincial Department of Education under Grant Number 2022AH051864.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare there are no conflicts of interest.
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
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
Sun, Z., Liu, X., Zhu, S. et al. Negacyclic BCH codes of length \(\frac{q^{2m}-1}{q+1}\) and their duals. Des. Codes Cryptogr. (2024). https://doi.org/10.1007/s10623-024-01380-3
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10623-024-01380-3