Type IV codes over a non-local non-unital ring

Authors

DOI:

https://doi.org/10.22199/issn.0717-6279-2020-04-0060

Keywords:

Non-unital rings, Semi-local rings, Self-orthogonal codes, Type IV codes, Modular lattices

Abstract

There is a local ring H of order 4, without identity for the multiplication, defined by generators and relations as H =?a, b | 2a = 2b = 0, a2 = 0, b2 = b, ab = ba = 0?. We classify self orthogonal codes of length n and size 2n (called here quasi self-dual codes or QSD) up to the length n = 6. In particular, we classify quasi Type IV codes (a subclass of Type IV codes, viz QSD codes with even weights) up to n = 6.

Author Biographies

Adel Alahmadi, King Abdulaziz University.

Dept. of Mathematics.

Amani Alkathiry, Umm Al-Qura University.

Dept. of Mathematics.

Alaa Altassan, King Abdulaziz University.

Dept. of Mathematics.

Widyan Basaffar, King Abdulaziz University.

Dept. of Mathematics.

Alexis Bonnecaze, Aix-Marseille Université.

CNRS.

Hatoon Shoaib, King Abdulaziz University.

Dept. of Mathematics.

Patrick Solé, Aix-Marseille Université.

CNRS.

References

A. Alahmadi, A. Altassan, W. Basaffar, A. Bonnecaze, H. Shoaib, and P. Solé, “Quasi Type IV codes over a non-unital ring”, submitted preprint, 2020. [On line]. Available: https://bit.ly/3gozCvo

A. Alahmadi, A. Alkathiry, A. Altassan, A. Bonnecaze, H. Shoaib, and P. Solé, “The build-up construction of quasi self-dual codes over a commutative non-unital ring”, submitted preprint, 2020.

A. Alahmadi, A. Alkathiry, A. Altassan, A. Bonnecaze, H. Shoaib, and P. Solé, “The build-up construction of quasi self-dual codes over a non-unital ring”, Journal of algebra and applications, accepted preprint, 2020. [On line]. Available: https://bit.ly/38HMJ8t

A. Alahmadi, A. Altassan, W. Basaffar, A. Bonnecaze, H. Shoaib, and P. Solé, “Type IV codes over a non-unital ring”, Journal of algebra and applications, accepted preprint, 2020. [On line]. Available: https://bit.ly/3f5D8ue

C. Bachoc, “Application of coding theory to the construction of modular lattices”, Journal of combinatorial theory, Series A, vol. 78, no. 1, pp. 92-119, Apr. 1997, doi:10.1006/jcta.1996.2763

A. Betten, M. Braun, H. Fripertinger, A. Kerber, A. Kohnert and A. Wassermann, Error correcting-linear codes. Berlin: Springer, 2006, doi: 10.1007/3-540-31703-1

Computational Algebra Group. and University of Sydney, “Magma computational algebra system”, Magma computer algebra, 2010. [Online]. Available: http://magma.maths.usyd.edu.au/magma/

J. H. Conway and N. J. A. Sloane, Sphere packings, lattices and groups, 2nd ed. New York, NY: Springer, 2003, doi: 10.1007/978-1-4757-2249-9

S. T. Dougherty, P. Gaborit, M. Harada, A. Munemasa, and P. Solé, “Type IV self-dual codes over rings”, IEEE transactions on information theory, vol. 45, no. 7, pp. 2345-2360, Nov. 1999, doi:10.1109/18.796375

B. Fine, “Classification of Finite Rings of Order p2,” Mathematics magazine, vol. 66, no. 4, pp. 248–252, Oct. 1993, doi: 10.1080/0025570X.1993.11996133

H. Fripertinger, “Enumeration of isometry classes of linear (n, k)-codes over GF (q), SYMMETRICA”, Bayreuther mathematische schriften, vol. 49, pp. 215-223, 1999. [On line]. Available: https://bit.ly/31QmYRC

X. D. Hou, “On the number of inequivalent binary self-orthogonal codes”, IEEE transactions on information theory, vol. 53, no. 7, pp. 2459-2479, Jul. 2007, doi: 10.1109/TIT.2007.899542

J.-L. Kim and Y. Lee, “Euclidean and Hermitian self-dual MDS codes over large finite fields”, Journal of combinatorial theory, series A, vol. 105, no. 1, pp. 79–95, Jan. 2004, doi: 10.1016/j.jcta.2003.10.003

J-L. Kim and Y. Lee, “An efficient construction of self-dual codes”, Bulletin Korean Mathematical Society, vol. 52, no. 3, pp. 915-923, 2015, doi: 10.4134/BKMS.2015.52.3.915

F. J. MacWilliams, N. J. A. Sloane, Eds., The theory of error-correcting codes, North-Amsterdam: North-Holland, 1977, doi: 10.1016/s0924-6509(08)x7030-8

G. Nebe and N. Sloane, “A catalogue of lattices,” Index to catalogue of lattices. [Online]. Available: http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/

Y-H. Park, “The classification of self-dual modular codes”, Finite fields and their applications, vol. 17, no. 5, pp. 442-460, Sep. 2011, doi: 10.1016/j.ffa.2011.02.010

Published

2020-07-28

How to Cite

[1]
A. Alahmadi, “Type IV codes over a non-local non-unital ring”, Proyecciones (Antofagasta, On line), vol. 39, no. 4, pp. 963-978, Jul. 2020.

Most read articles by the same author(s)