Abstract
Let \({\mathfrak {R}}={\mathbb {F}}_{p^m}+u{\mathbb {F}}_{p^m}\) with \(u^2=0\), where m, s are positive integers and p is an odd prime. For any invertible element \(\varLambda\) of \({\mathfrak {R}}\), the symbol-pair distances of all \(\varLambda\)-constacyclic codes of length \(2p^s\) over \({\mathfrak {R}}\) are completely obtained. We identify all symbol-pair Maximum Distance Separable (MDS) constacyclic codes of length \(2p^s\) over \({\mathfrak {R}}\). As examples, many new symbol-pair codes, as well as symbol-pair MDS codes are constructed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Berman, S.D.: Semisimple cyclic and Abelian codes. II. Kibernetika (Kiev) 3, 21–30 (1967) (Russian). English translation: Cybernetics 3, 17–23 (1967)
Cassuto, Y., Blaum, M.: Codes for symbol-pair read channels. IEEE Trans. Inf. Theory 57(12), 8011–8020 (2011)
Chen, B., Dinh, H.Q., Liu, H., Wang, L.: Constacyclic codes of length \(2p^s\) over \({\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}\). Finite Fields Appl. 37, 108–130 (2016)
Chen, B., Lin, L., Liu, H.: Constacyclic symbol-pair codes: lower bounds and optimal constructions. IEEE Trans. Inf. Theory 12, 7661–7666 (2017)
Chee, Y.M., Ji, L., Kiah, H.M., Wang, C., Yin, J.: Maximum distance separable codes for symbol-pair read channels. IEEE Trans. Inf. Theory 59(11), 7259–7267 (2013)
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., Yamaka, W.: MDS constacyclic codes and mds symbol-pair constacyclic codes. IEEE Access 9, 137970–137990 (2021)
Dinh, H.Q., Dhompongsa, S., Sriboonchitta, S.: On constacyclic codes of length \(4p^s\) over \(F_{p^m}+ uF_{p^m}\). Discrete Math. 340(4), 832–849 (2017)
Dinh, H.Q.: On the linear ordering of some classes of negacyclic and cyclic codes and their distance distributions. Finite Fields Appl. 14(1), 22–40 (2008)
Dinh, H.Q., López-Permouth, S.R.: Cyclic and negacyclic codes over finite chain Rings. IEEE Trans. Inform. Theory 50, 1728–1744 (2004)
Dinh, H.Q., Nguyen, B.T., Singh, A.K., Sriboonchitta, S.: Hamming and symbol-pair distances of repeated-root constacyclic codes of prime power lengths over \({\mathbb{F}}_ {p^ m}+ u{\mathbb{F}}_ p^ m\). IEEE Commun. Lett. 22(12), 2400–2403 (2018)
Dinh, H.Q., Wang, X., Liu, H., Sriboonchitta, S.: On the symbol-pair distances of repeated-root constacyclic codes of length \(2p^s\). Discrete Math. 342(11), 3062–3078 (2019)
Dinh, H.Q., Nguyen, B.T., Sriboonchitta, S.: MDS symbol-pair cyclic codes of length \(2p^s\) over \({\mathbb{F}}_{p^m}\). IEEE Trans. Inf. Theory 66(1), 240–262 (2020)
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 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. 19(11), 2050209 (2019)
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., Singh, A.K., Thakur, M.K.: On Hamming and b-symbol distance distributions of repeated-root constacyclic codes of length \(4p^s\) over \({\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}\). J. Appl. Math. Comput. 66, 885–905 (2021)
Ding, B., Ge, G., Zhang, J., Zhang, T., Zhang, Y.: New constructions of mds symbol-pair codes. Des. Codes Cryptogr. 4, 841–859 (2018)
Falkner, G., Kowol, B., Heise, W., Zehendner, E.: On the existence of cyclic optimal codes. Atti Sem. Mat. Fis. Univ. Modena 28, 326–341 (1979)
Guenda, K., Gulliver, T.A.: Repeated-root constacyclic codes of length \(mp^s\) over \({\mathbb{F}}_{p^r}+ u{\mathbb{F}}_{p^r}...+u^{e-1}{\mathbb{F}}_{p^r}\). J. Algebra Appl. 14(01), 1450081 (2015)
Kai, X., Zhu, S., Li, P.: A construction of new MDS symbol-pair codes. IEEE Trans. Inf. Theory 61(11), 5828–5834 (2015)
López-Permouth, S.R., Özadam, H., Özbudak, F., Szabo, S.: Polycyclic codes over Galois rings with applications to repeated-root constacyclic codes. Finite Fields Appl. 19(1), 16–38 (2013)
Massey, J.L., Costello, D.J., Justesen, J.: Polynomial weights and code constructions. IEEE Trans. Inf. Theory 19(1), 101–110 (1973)
Roth, R., Seroussi, G.: On cyclic mds codes of length \(q\) over \(GF (q)\). IEEE Trans. Inf. Theory 32(2), 284–285 (1986)
Acknowledgements
The authors are thankful for the research support from the INSPIRE Programme (INSPIRE Code-IF160375) of DST, Govt. Of India and IIT(ISM), Dhanbad, India.
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.
Authors’ names are in alphabetical order. This is a part of Ph.D. work of Mr. M. K. Thakur.
Rights and permissions
About this article
Cite this article
Dinh, H.Q., Singh, A.K. & Thakur, M.K. On symbol-pair distances of repeated-root constacyclic codes of length \(2p^s\) over \({\mathbb {F}}_{p^m}+u{\mathbb {F}}_{p^m}\) and MDS symbol-pair codes. AAECC 34, 1027–1043 (2023). https://doi.org/10.1007/s00200-021-00534-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00200-021-00534-3