Coding Matrices for $GL(2,q)$

Year 2018, Volume: 1 Issue: 2, 118 - 130, 25.12.2018

Ahmed A. Khammash Marwa M. Hamed

https://doi.org/10.33401/fujma.462055

Cited By: 1

Abstract

We use the BN-pair structure for the general linear group to write a suitable listing of the elements of the finite group $GL(2,q)$ which is then used to determine its ring of matrices. This approach of identifying finite group ring with ring of matrices has been used effectively to construct linear codes, benefiting from the ring-theoretic structure of both group rings and the ring of matrices.

Keywords

Group ring, G-matrix, BN-pair, Error-correcting codes

References

• [1] F. J. Macwilliams, Codes and ideals in group algebras, Comb. Math. Appl., (1969), 312-328.
• [2] S. D. Berman, On the theorey of group codes, Kibernetika, 3(1) (1967), 31-39.
• [3] R. Ferraz, Polcino Milies, Idempotents in group algebras and minimal abelian codes, Finite Fields Appl., 13 (2007), 382-393.
• [4] T. Hurley, Group rings and ring of matrices, Inter. J. Pure Appl. Math., 31(3) (2006), 319-335.
• [5] P. Hurley, T. Hurely, Block Codes from Matrix and Group Rings, Chapter 5, (Eds.) I. Woungang, S. Misra, S.C. Misma, Selected Topics in Information and Coding Theory, World Scientific, 2010, 159-194.
• [6] C. Curtis, I. Reiner, Methods of Representation Theory, Wiley, New York, 1987.