Abstract
There is a local ring I of order 4, without identity for the multiplication, defined by generators and relations as
We study a recursive construction of self-orthogonal codes over I. We classify self orthogonal codes of length n and size \(2^n\) (called here quasi self-dual codes or QSD) up to the length \(n=6.\) In particular, we classify Type IV codes (QSD codes with even weights) and quasi Type IV codes (QSD codes with even torsion code) up to \(n=6.\)
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alahmadi A., Alkathiry A., Altassan A., Bonnecaze A., Shoaib H., Solé P.: The build-up construction of quasi self-dual codes over a non-unital ring. J. Algebra Appl. (2020). https://doi.org/10.1142/S0219498822501432.
Alahmadi A., Altassan A., Basaffar W., Bonnecaze A., Shoaib H., Solé P.: Type IV codes over a non-unital ring. J. Algebra Appl. (2021). https://doi.org/10.1142/S0219498822501420.
Alahmadi A., Altassan A., Basaffar W., Bonnecaze A., Shoaib H., Solé P.: Quasi Type IV codes over a non-unital ring. J. AAECC 32, 217–228 (2021).
Conway J.H., Sloane N.J.A.: Self-dual codes over the integers modulo four. J. Comb. Theory Ser. A 62, 30–45 (1993).
Dougherty S., Leroy A.: Euclidean self-dual codes over non-commutative Frobenius rings. Appl. Algebra Eng. Commun. Comput. 27(3), 185–203 (2016).
Dougherty S.T., Gaborit P., Harada M., Munemasa A., Solé P.: Type IV self-dual codes over rings. IEEE Trans. Inf. Theory 45(7), 2345–2360 (1999).
Fine B.: Classification of finite rings of order \(p^2\). Math. Mag. 66(4), 248–252 (1993).
Gaborit P.: Mass formulas for self-dual codes over \( {\mathbb{Z}}_4\) and \({\mathbb{F}}_q+u{\mathbb{F}}_q\) rings. IEEE Trans. Inf. Theory 42, 1222–1228 (1996).
Hammons A.R. Jr., Kumar P.V., Calderbank A.R., Sloane N.J.A.: Patrick Solé: The \(Z_4\)-linearity of Kerdock, Preparata, Goethals, and related codes. IEEE Trans. Inf. Theory 40(2), 301–319 (1994).
Han S., Lee H., Lee Y.: Construction of self-dual codes over \({\mathbb{F}}_2+u{\mathbb{F}}_2,\) Bull. Korean Math Soc. 49, 135–143 (2012).
Hou X.D.: On the number of inequivalent binary self-orthogonal codes. Trans. Inf. Theory 53, 2459–2479 (2007).
Hufman W.C., Pless V.: Fundamentals of Error Correcting Codes. Cambridge University Press, Cambridge (2003).
Kim J.-L., Lee Y.: Euclidean and Hermitian self-dual MDS codes over large finite fields. J. Comb. Theory A 105, 79–95 (2004).
Kim J.-L., Lee Y.: An efficient construction of self-dual codes. Bull. Korean Math. Soc. 52, 915–923 (2015).
MacWilliams F.J., Sloane N.J.A.: The Theory of Error-Correcting Codes. North-Holland, Amsterdam (1977).
Raghavendran R.: A class of finite rings. Compos. Math. 21, 195–229 (1969).
Shi M., Alahmadi A., Solé P.: Codes and Rings: Theory and Practice. Academic Press, Cambridge (2017).
Solé P.: Codes over Rings. World Scientific, Singapore (2008).
Acknowledgements
This project was funded by the Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah, under Grant Number KEP-PhD-29-130-40 . The authors, therefore, acknowledge with thanks DSR for technical and financial support.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by J.-L. Kim.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue: On Coding Theory and Combinatorics: In Memory of Vera Pless”.
Rights and permissions
About this article
Cite this article
Alahmadi, A., Alkathiry, A., Altassan, A. et al. The build-up construction over a commutative non-unital ring. Des. Codes Cryptogr. 90, 3003–3010 (2022). https://doi.org/10.1007/s10623-022-01044-0
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10623-022-01044-0