Abstract
In this paper, we give the exact number of \({{\mathbb {Z}}}_{2}{{\mathbb {Z}}}_{4}\)-additive cyclic codes of length \(n=r+s,\) for any positive integer r and any positive odd integer s. We will provide a formula for the the number of separable \({{\mathbb {Z}}_{2}{{\mathbb {Z}}_{4}}}\)-additive cyclic codes of length n and then a formula for the number of non-separable \({{\mathbb {Z}} _{2}{{\mathbb {Z}}_{4}}}\)-additive cyclic codes of length n. Then, we have generalized our approach to give the exact number of \({{\mathbb {Z}}_{p}{\mathbb { Z}_{p^{2}}}}\)-additive cyclic codes of length \(n=r+s,\) for any prime p, any positive integer r and any positive integer s where \(\gcd \left( p,s\right) =1.\) Moreover, we will provide examples of the number of these codes with different lengths \(n=r+s\).
Similar content being viewed by others
References
Abualrub, T., Siap, I., Aydin, N.: \({\mathbb{Z}}_{2}{\mathbb{Z}} _{4}\)-additive cyclic codes. IEEE Trans. Inf. Theory 60(3), 1508–1514 (2014)
Aydogdu, I., Abualrub, T., Siap, I.: \({\mathbb{Z}}_{2} {\mathbb{Z}}_{2}[u]\)-cyclic and constacyclic codes. IEEE Trans. Inf. Theory 63(8), 4883–4893 (2017)
Borges, J., Fernández-Córdoba, C., Pujol, J., Rif à, J., Villanueva, M.: \(\mathbb{Z}_{2}\mathbb{Z}_{4}\)-linear codes: generator matrices and duality. Des. Codes Cryptogr. 54(2), 167–179 (2010)
Borges, J., Fernández-Córdoba, C., Ten-Valls, R.: \(\mathbb{Z}_{2}\mathbb{Z}_{4}\)-additive cyclic codes, generator polynomials and dual codes. IEEE Trans. Inf. Theory 62(11), 6348–6354 (2016)
Borges, J., Fernández-Córdoba, C., Ten-Valls, R.: On \(\mathbb{Z}_{p^r}\mathbb{Z}_{p^s}\)-additive cyclic codes. Adv. Math. Commun. 12(1), 169–179 (2018)
Carlet, C.: \({\mathbb{Z}}_{2^{k}}\)-linear codes. IEEE Trans. Inf. Theory 44, 1543–1547 (1998)
Dougherty, S., Salturk, E.: Counting additive \({\mathbb{Z}} _{2}{\mathbb{Z}}_{4}\) codes. Contemp. Math. 634, 137–147 (2015)
Greferath, M., Schmidt, S.E.: Gray isometries for finite chain rings. IEEE Trans. Inf. Theory 45(7), 2522–2524 (1999)
Hammons, A.R., Kumar, P.V., Calderbank, A.R., Sloane, N.J.A., Solé, P.: The \({\mathbb{Z}}_{4}\)-linearity of Kerdock, Preparata, Goethals and related codes. IEEE Trans. Inf. Theory 40(2), 301–319 (1994)
Honold, T., Landjev, I.: Linear codes over finite chain rings. In: In Optimal Codes and Related Topics, Sozopol, Bulgaria, pp. 116–126 (1998)
Rifà-Pous, H., Rifà, J., Ronquillo, L.: \({\mathbb{Z}} _{2}{\mathbb{Z}}_{4}\)-additive perfect codes in steganography. Adv. Math. Commun. 5(3), 425–433 (2011)
Siap, I., Aydogdu, I.: Counting the generator matrices of \({ \mathbb{Z}}_{2}{\mathbb{Z}}_{8}\) codes. Math. Sci. Appl. E-Notes 1(2), 143–149 (2013)
Acknowledgements
The authors would like to thank the reviewers for their careful reading of the paper and their valuable comments that improved the paper tremendously. In fact, the reviewers comments played a huge impact to create Sect. 5 of the paper that generalizes our results from counting the number of \({ {\mathbb {Z}}_{2}{{\mathbb {Z}}_{4}}}\)-additive cyclic codes to counting the numbers of \({{\mathbb {Z}}_{p}{{\mathbb {Z}}_{p^{2}}}}\)-additive cyclic codes.
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
Yildiz, E., Abualrub, T. & Aydogdu, I. On the number of \({{\mathbb {Z}}}_{2}{{\mathbb {Z}}}_{4}\) and \({{\mathbb {Z}}}_{p}{{\mathbb {Z}}}_{p^{2}}\)-additive cyclic codes. AAECC 34, 81–97 (2023). https://doi.org/10.1007/s00200-020-00474-4
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00200-020-00474-4
Keywords
- \({{\mathbb {Z}}}_{2}{{\mathbb {Z}}}_{4}\)-additive cyclic codes
- \({{\mathbb {Z}}}_{p}{{\mathbb {Z}}}_{p^{2}}\)-additive cyclic codes
- counting
- separable
- non-separable codes