Abstract
Let p be an odd prime, and \(\delta\) be an arbitrary unit of the finite chain ring \({\mathbb {F}}_{p^m}+u{\mathbb {F}}_{p^m} \,\, (u^2=0)\). The Hamming distances of all \(\delta\)-constacyclic codes of length \(2p^s\) over \({\mathbb {F}}_{p^m}+u{\mathbb {F}}_{p^m}\) are completely determined. We provide some examples from which some codes have better parameters than the existing ones. As applications, we determine all MDS repeated-root \(\delta\)-constacyclic codes of length \(2p^s\) over \(\mathbb F_{p^m}+u{\mathbb {F}}_{p^m}\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Cao, Y.-L., Cao, Y., Dinh, H.Q., Fu, F.-W., Gao, J., Sriboonchitta, S.: Constacyclic codes of length \(np^s\) over \({\mathbb{F}}_{p^m} + u{\mathbb{F}}_{p^m}\). Adv. Math. Commun. 12, 231–262 (2018)
Cao, Y.-L., Cao, Y., Dinh, H.Q., Fu, F.-W., Gao, J., Sriboonchitta, S.: A class of repeated-root constacyclic codes over \({\mathbb{F}}_{p^m}[u]/\langle u^e \rangle\) of Type \(2\). Finite Fields Appl. 55, 238–267 (2019)
Cao, Y.-L., Cao, Y., Dinh, H.Q., Jitman, S.: An explicit representation and enumeration for self-dual cyclic codes over \({\mathbb{F}}_{2^m}+u{\mathbb{F}}_{2^m}\) of length \(2^s\). Discrete Math. 342, 2077–2091 (2019)
Cao, Y.-L., Cao, Y., Dinh, H.Q., Fu, F.-W., Gao, J., Sriboonchitta, S.: A class of linear codes of length \(2\) over finite chain rings. J. Algebra Appl. 19, 2050103.1-15 (2020)
Cao, Y.-L., Cao, Y., Dinh, H.Q., Fu, F.-W., Ma, F.: Construction and enumeration for self-dual cyclic codes of even length over \({\mathbb{F}}_{2^m} + u{\mathbb{F}}_{2^m}\). Finite Fields Appl. 61, (2020). (in press)
Cao, Y.-L., Cao, Y., Dinh, H.Q., Fu, F.-W., Maneejuk, P.: On matrix-product structure of repeated-root constacyclic codes over finite fields. Discrete Math. 343, 111768 (2020). in press
Cao, Y.-L., Cao, Y., Dinh, H.Q., Bandi, R., Fu, F.-W.: An explicit representation and enumeration for negacyclic codes of length \(2^kn\) over \({\mathbb{Z}}_4+u{\mathbb{Z}}_4\). Adv. Math. Commun. (2020). (in press)
Cao, Y.-L., Cao, Y., Dinh, H.Q., Jitman, S.: An efficient method to construct self-dual cyclic codes of length \(p^s\) over \({\mathbb{F}}_{p^m} + u{\mathbb{F}}_{p^m}\). Discrete Math. 343, 111868 (2020). in press
Cao, Y.-L., Cao, Y., Dinh, H.Q., Bag, T., Yamaka, W.: Explicit representation and enumeration of repeated-root \((\delta +\alpha u^2)\)-constacyclic codes over \({\mathbb{F}}_{2^m}[u]/\langle u^{2\lambda }\rangle\). IEEE Access 8, 55550–55562 (2020)
Chen, B., Dinh, H.Q., Liu, H., Wang, L.: Constacyclic codes of length \(2p^s\) over\({\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}\). J. Finite Fields Appl. 37(Issue C), 108–130 (2016)
Castagnoli, G., Massey, J.L., Schoeller, P.A., von Seemann, N.: On repeated-root cyclic codes. IEEE Trans. Inf. Theory 37, 337–342 (1991)
Dinh, H.Q.: On repeated-root constacyclic codes of length \(4p^s\). Asian Eur. J. Math. 6, 1–25 (2013)
Dinh, H.Q.: Repeated-root constacyclic codes of length \(2p^s\). Finite Fields Appl. 18, 133–143 (2012)
Dinh, H.Q.: Structure of repeated-root constacyclic codes of length \(3p^s\) and their duals. Discrete Math. 313, 983–991 (2013)
Dinh, H.Q.: On the linear ordering of some classes of negacyclic and cyclic codes and their distance distributions. Finite Fields Appl. 14, 22–40 (2008)
Dinh, H.Q.: Constacyclic codes of length \(p^s\) over \({\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}\). J. Algebra 324, 940–950 (2010)
Dinh, H.Q., Nguyen, B.T., Singh, A.K., Sriboonchitta, S.: Hamming and symbol pair distances of repeated root constacycliccodes of prime power lengths over \({\mathbb{F}}\_{p^m} +u{\mathbb{F}}\_{p^m}\). IEEE Trans. Inf. Theory 64(4), 2417–2430 (2018)
Dinh, H.Q., Wang, X., Liu, H., Sriboonchitta, S.: On the Hamming distances of repeated-root constacyclic codes of length \(4p^s\). Discrete Math. 342(5), 1456–1470 (2019)
Dinh, H.Q., Nguyen, B.T., Singh, A.K., Sriboonchitta, S.: On the symbol-pair distance of repeated-root constacyclic codes of prime power lengths. IEEE Trans. Inf. Theory 64(4), 2417–2430 (2018)
Dinh, H.Q., Satpati, S., Singh, A.K., Yamaka, W.: Symbol-triple distance of repeated-root constacylic codes of prime power lengths. J. Algebra Appl. (2019). https://doi.org/10.1142/S0219498820502096
Dinh, H.Q., Kumam, P., Kumar, P., Satpati, S., Singh, A.K., Yamaka, W.: MDS symbol-pair repeated-root constacylic codes of prime power lengths over \({\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}\). IEEE Access 7, 145039–145048 (2019)
Dinh, H.Q., Wang, X., Sirisrisakulchai, J.: Optimal b-symbol constacyclic codes with respect to the singleton bound. J. Algebra Appl. (2020). https://doi.org/10.1142/S0219498820501510
Dinh, H.Q., Wang, X., Sirisrisakulchai, J.: On the Hamming distance of constacyclic codes of length \(5p^s\). IEEE Access 8, 46242–46254 (2020)
Dinh, H.Q., Wang, X., Maneejuk, P.: On the Hamming distance of repeated-root cyclic codes of length \(6p^s\). IEEE Access 8, 39946–39958 (2020)
Grassl, M.: Bounds on the minimum distance of linear codes and quantum codes. Online available at http://www.codetables.de. Accessed 7 May 2019
Ling, S., Xing, C.: Coding Theory—A First Course. Cambridge Publications, Cambridge (2004)
Liu, Y., Shi, M., Dinh, H.Q., Sriboonchitta, S.: Repeated-root constacyclic codes of length \(3\ell ^mp^s\), Advances Math. Comm. (2019) (to appear)
Lopez-Permouth, S.R., Ozadam, H., Ozbudak, F., Szabo, S.: Polycyclic codes over Galois rings with applications to repeated-root constacyclic codes. Finite Fields Appl. 19, 16–38 (2013)
Ozadam, H., Ozbudak, F.: The minimum hamming distance of cyclic codes of length \(2p^s\). Appl. Algebra Eng. Commun. Comput. LNCS 5527, 92–100 (2009)
van Lint, J.H.: Repeated-root cyclic codes. IEEE Trans. Inform. Theory 37, 343–345 (1991)
Acknowledgements
The authors would like to sincerely thank the referees and the editor for a very meticulous reading of this manuscript, and for valuable suggestions which help to create an improved final version.This paper is partially supported by the Centre of Excellence in Econometrics, Faculty of Economics, Chiang Mai University.
Author information
Authors and Affiliations
Corresponding author
Additional information
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
Dinh, H.Q., Gaur, A., Gupta, I. et al. Hamming distance of repeated-root constacyclic codes of length \(2p^s\) over \({\mathbb {F}}_{p^m}+u{\mathbb {F}}_{p^m}\). AAECC 31, 291–305 (2020). https://doi.org/10.1007/s00200-020-00432-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00200-020-00432-0