Abstract
We give a new technique for constructing self-dual codes based on a block matrix whose blocks arise from group rings and orthogonal matrices. The technique can be used to construct self-dual codes over finite commutative Frobenius rings of characteristic 2. We give and prove the necessary conditions needed for the technique to produce self-dual codes. We also establish the connection between self-dual codes generated by the new technique and units in group rings. Using the construction together with the building-up construction, we obtain new extremal binary self-dual codes of lengths 64, 66 and 68 and new best known binary self-dual codes of length 80.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data availability
Not applicable.
Code availability
Not applicable.
References
Betsumiya K., Georgiou S., Gulliver T.A., Harada M., Koukouvinos C.: On self-dual codes over some prime fields. Discret. Math. 262(1–3), 37–58 (2003). https://doi.org/10.1016/S0012-365X(02)00520-4.
Bouyukliev I.G.: What is Q-extension? Serdica J. Comput. 1(2), 115–130 (2007).
Buyuklieva S., Boukliev I.: Extremal self-dual codes with an automorphism of order 2. IEEE Trans. Inf. Theory 44(1), 323–328 (1998). https://doi.org/10.1109/18.651059.
Conway J.H., Sloane N.J.A.: A new upper bound on the minimal distance of self-dual codes. IEEE Trans. Inf. Theory 36(6), 1319–1333 (1990). https://doi.org/10.1109/18.59931.
Dougherty S.T., Gulliver T.A., Harada M.: Extremal binary self-dual codes. IEEE Trans. Inf. Theory 43(6), 2036–2047 (1997). https://doi.org/10.1109/18.641574.
Dougherty S.T., Gaborit P., Harada M., Solé P.: Type II codes over \(\mathbb{F} _2+u\mathbb{F} _2\). IEEE Trans. Inf. Theory 45(1), 32–45 (1999). https://doi.org/10.1109/18.746770.
Dougherty S.T., Kim J.-L., Kulosman H., Liu H.: Self-dual codes over commutative Frobenius rings. Finite Fields Appl. 16(1), 14–26 (2010). https://doi.org/10.1016/j.ffa.2009.11.004.
Dougherty S.T., Yildiz B., Karadeniz S.: Codes over \(R_k\), Gray maps and their binary images. Finite Fields Appl. 17(3), 205–219 (2011). https://doi.org/10.1016/j.ffa.2010.11.002.
Gaborit P., Pless V., Solé P., Atkin O.: Type II codes over \(\mathbb{F} _4\). Finite Fields Appl. 8(2), 171–183 (2002). https://doi.org/10.1006/ffta.2001.0333.
Gildea J., Kaya A., Taylor R., Yildiz B.: Constructions for self-dual codes induced from group rings. Finite Fields Appl. 51, 71–92 (2018). https://doi.org/10.1016/j.ffa.2018.01.002.
Gildea J., Kaya A., Korban A., Tylyshchak A.: Self-dual codes using bisymmetric matrices and group rings. Discret. Math. (2020). https://doi.org/10.1016/j.disc.2020.112085.
Gildea J., Kaya A., Yildiz B.: New binary self-dual codes via a variation of the four-circulant construction. Math. Commun. 25(2), 213–226 (2020).
Gildea J., Korban A., Kaya A., Yildiz B.: Constructing self-dual codes from group rings and reverse circulant matrices. Adv. Math. Commun. 15(3), 471–485 (2021). https://doi.org/10.3934/amc.2020077.
Gildea J., Korban A., Roberts A.M., Tylyshchak A.: Extremal binary self-dual codes from a bordered four circulant construction. Discret. Math. (2023). https://doi.org/10.1016/j.disc.2023.113425.
Gulliver T.A., Harada M.: On extremal double circulant self-dual codes of lengths 90–96. Appl. Algebra Eng. Commun. Comput. 30(5), 403–415 (2019). https://doi.org/10.1007/s00200-019-00381-3.
Harada M.: Binary extremal self-dual codes of length 60 and related codes. Des. Codes Cryptogr. 86(5), 1085–1094 (2018). https://doi.org/10.1007/s10623-017-0380-2.
Hurley T.: Group rings and rings of matrices. Int. J. Pure Appl. Math. 31(3), 319–335 (2006).
Ling S., Solé P.: Type II codes over \(\mathbb{F} _4+u\mathbb{F} _4\). Eur. J. Comb. 22(7), 983–997 (2001). https://doi.org/10.1006/eujc.2001.0509.
Mallows C.L., Sloane N.J.A.: An upper bound for self-dual codes. Inf. Control 22(2), 188–200 (1973). https://doi.org/10.1016/S0019-9958(73)90273-8.
Rains E.M.: Shadow bounds for self-dual codes. IEEE Trans. Inf. Theory 44(1), 134–139 (1998). https://doi.org/10.1109/18.651000.
Roberts A.M.: Weight enumerator parameter database for binary self-dual codes (2021). https://amr-wepd-bsdc.netlify.app.
Tsai H.-P.: Extremal self-dual codes of lengths 66 and 68. IEEE Trans. Inf. Theory 45(6), 2129–2133 (1999). https://doi.org/10.1109/18.782156.
Yankov N., Anev D., Gürel M.: Self-dual codes with an automorphism of order 13. Adv. Math. Commun. 11(3), 635–645 (2017). https://doi.org/10.3934/amc.2017047.
Funding
The author did not receive support from any organisation for the submitted work.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The author has no conflicts of interest to declare that are relevant to the content of this article.
Additional information
Communicated by J. H. Koolen.
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
Roberts, A.M. Self-dual codes from a block matrix construction characterised by group rings. Des. Codes Cryptogr. (2024). https://doi.org/10.1007/s10623-024-01359-0
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10623-024-01359-0