Abstract
In this study we consider Euclidean and Hermitian self-dual codes over the direct product ring \(\mathbb {F}_{2} \times (\mathbb {F}_{2}+v\mathbb {F}_{2})\) where v2 = v. We obtain some theoretical outcomes about self-dual codes via the generator matrices of free linear codes over \(\mathbb {F}_{2} \times (\mathbb {F}_{2}+v\mathbb {F}_{2})\). Also, we obtain upper bounds on the minimum distance of linear codes for both the Lee distance and the Gray distance. Moreover, we find some free Euclidean and free Hermitian self-dual codes over \(\mathbb {F}_{2} \times (\mathbb {F}_{2}+v\mathbb {F}_{2})\) via some useful construction methods.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Change history
22 December 2020
The original version of this article unfortunately contained some mistakes on the equations at page 4 of the published paper.
References
Aksoy, R., Çalışkan, F.: A new shortening method and Hermitian self-dual codes over \(\mathbb {F}_{2}+v\mathbb {F}_{2}\). Discrete Math. 343, 111716 (2020)
Aydogdu, I., Siap, I., Ten-Valls, R.: On the structure of \(\mathbb {Z}_{2} \mathbb {Z}_{2} [u^{3}]\)-linear and cyclic codes. Finite Fields Appl. 48, 241–260 (2017)
Bonnecase, A., Bracco, A.D., Dougherty, S.T., Nochefranca, L.R., Solé, P.: Cubic self-dual binary codes. IEEE Trans. Inform. Theory 49, 2253–2259 (2003)
Borges, J., Fernández-Córdoba, C., Ten-Valls, R.: Linear and cyclic codes over direct product of finite chain rings. Math. Methods Appl. Sci. 41, 6519–6529 (2018)
Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system I: The user language. J. Symbolic Comput. 24, 235–265 (1997)
Cengellenmis, Y., Dertli, A., Aydin, N.: Some constacyclic codes over \(\mathbb Z_{4}\left [u\right ]/\left \langle u^{2}\right \rangle \), new Gray maps, and new quaternary codes. Algebra Colloq. 25, 369–376 (2018)
Çalışkan, F., Aksoy, R.: Linear codes over \(\mathbb {F}_{2} \times (\mathbb {F}_{2}+v\mathbb {F}_{2})\) and the MacWilliams identities. Appl. Algebra Engrg. Comm. Comput. 31, 135–147 (2020)
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. Inform. Theory 40, 301–319 (1994)
Harada, M.: Binary extremal self-dual codes of length 60 and related codes. Des. Codes Cryptogr. 86, 1085–1094 (2018)
Huffman, W.C.: On the classification and enumeration of self-dual codes. Finite Fields Appl. 11, 451–490 (2005)
Karadeniz, S., Aksoy, R.: Self-dual Rk lifts of binary self-dual codes. Finite Fields Appl. 34, 317–326 (2015)
Karadeniz, S., Dougherty, S.T., Yildiz, B.: Constructing formally self-dual codes over Rk. Discrete Appl. Math. 167, 188–196 (2014)
Kaya, A., Yildiz., B., Siap, I.: New extremal binary self-dual codes from \(\mathbb {F}_{4}+u\mathbb {F}_{4}\)-lifts of quadratic circulant codes over \(\mathbb {F}_{4}\). Finite Fields Appl. 35, 318–329 (2015)
Kim, J.L., Lee, Y.: An efficient construction of self-dual codes. Bull. Korean Math. Soc. 52, 915–923 (2015)
Rains, E.M., Sloane, N.J.A.: Self-dual codes. In: Pless, V., Huffman, W.C. (eds.) The Handbook of Coding Theory. North-Holland, New York (1998)
Yildiz, B., Karadeniz, S.: Self-dual codes over \(\mathbb {F}_{2}+u\mathbb {F}_{2}+v\mathbb {F}_{2}+uv\mathbb {F}_{2}\). J. Franklin Inst. 347, 1888–1894 (2010)
Acknowledgements
The authors would like to thank the anonymous reviewers and the editor for their valuable comments which significantly improved the original manuscript.
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.
This work was supported by Scientific Research Projects Coordination Unit of Istanbul University (Project No 29539).
Rights and permissions
About this article
Cite this article
Aksoy, R., Çalışkan, F. Self-dual codes over \(\mathbb {F}_{2} \times (\mathbb {F}_{2}+v\mathbb {F}_{2})\). Cryptogr. Commun. 13, 129–141 (2021). https://doi.org/10.1007/s12095-020-00461-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12095-020-00461-z