Coding Matrices for the Semi-Direct Product Groups

Year 2020, , 109 - 115, 15.12.2020

Amnah Alkinani , Ahmed Khammash

https://doi.org/10.33401/fujma.690424

Abstract

We shall determine the coding matrix of the semi-direct product group $ G = C_{n} \rtimes_{\phi} C_{m} $ ; $ \phi : C_{m} \longrightarrow Aut(C_{n}) $ of two cyclic groups in order to generalize the known result for the dihedral group $D_{2n}$, which is known to be a semi-direct of the two cyclic groups $C_{n} \ , \ C_{2}$.

Keywords

Code, Group ring, Ring of matrices, Semi-direct product group

References

• [1] R. Hamming, Error detecting and error correcting codes, The Bell Syst. Tech. J., 29 (1950), 147-160.
• [2] F. J. MacWilliams, Codes and ideals in group algebra, Comb. Math. Appl., (1969), 317-328.
• [3] T. Hurley, Group rings and rings of matrices, Int. J. Pure Appl. Math, 31(3) (2006), 319-335.
• [4] P. Hurley, T. Hurley, Codes from zero-divisors and units in group rings, (2007), arXiv:0710.5893v1 [cs.IT].
• [5] M. Hamed, Constructing codes from group rings, Msc dissertation, Umm Al-Qura University, 2018.
• [6] M. Hamed, A. Khammash, Coding matrices for GL (2, q), Fundam. J. Math. Appl., 1(2) (2018), 118-130.
• [7] P. Hurley, T. Hurley, Block codes from matrix and group rings, Chapter 5, in Selected topics in information and coding theory, I. Woungang, S. Misra, (Eds.), World Scientific, (2010), 159-194.