On some Ternary Linear Codes and Designs obtained from the Projective Special Linear Goup PSL3(4)

Main Article Content

Yvonne W. Kariuki
Ojiema M. Onyango
Okombo M. Immaculate

Abstract

Let (G, ∗) be a group and X any set, an action of a group G on X, denoted as G×X → X, (g, x) 7→ g.x, assigns to each element in G a transformation of X that is compatible with the group structure of G. If G has a subgroup H then there is a transitive group action of G on the set (G/H) of the right co-sets of H by right multiplication. A representation of a group G on a vector space V carries the dimension of the vector space. Now, given a field F and a finite group G, there is a bijective correspondence between the representations of G on the finitedimensional F-vector spaces and finitely generated FG-modules. We use the FG -modules to construct linear ternary codes and combinatorial designs from the permutation representations of the group L3(4). We investigate the properties and parameters of these codes and designs.

We further obtain the lattice structures of the sub-modules and compare these ternary codes with the binary codes constructed from the same group.

Keywords:
Ternary linear codes, simple groups, designs, linear groups.

Article Details

How to Cite
Kariuki, Y. W., Onyango, O. M., & Immaculate, O. M. (2019). On some Ternary Linear Codes and Designs obtained from the Projective Special Linear Goup PSL3(4). Asian Research Journal of Mathematics, 15(4), 1-17. https://doi.org/10.9734/arjom/2019/v15i430152
Section
Original Research Article

References

Hill R, Newton DE. Optimal ternary linear codes. Designs, Codes and Cryptography. 1992;2(2):137-157.

Hamming RH. Error detecting and error correcting codes. The Bell System Technical Journal. 1950;XXIX(2).

Brooke PHL. On the Steiner system S(2,4,28) and codes associated with the simple group of order 6048. J. Algebra. 1985;97:376-406.

Key JD, Moori J. Designs, codes and graphs from the janko groups J1 and J2. J. Combin.
Math. Combin. Comput. 2002;40:143-159.

Moori J, Rodrigues BG. A self-orthogonal doubly even code invariant under McL :2. J. Combin.
Theory Ser. A. 2005;110(1):5369.

Blake IF, Cohen G, Deza M. Coding with permutations. Information and Control. 1979;43:1-19.

Pace Nicola. New ternary linear codes from projectivity groups. Discrete Mathematics. 2014;331:22-26.

Chikamai L, Moori J, Rodrigues BG. Linear codes obtained from 2-modular representations of somefinite simple groups. Ph.D, Thesis, University of Kwazulu-Natal; 2012.

Conway J.H; Curtis RT, S. P, Norton SP, Parker RA, Wilson RA. Atlas of finite groups: Maximal subgroups and ordinary characters for simple groups. Clarendon Press; 1985.

Gorenstein D, Lyons R, Solomon R. The classification of the finite simple groups. Amer. Math. Soc.,Providence RI; 1994.

Anderson Ian. A first course in combinatorial mathematics. Oxford University Press. 2nd edition; 1989.

Cameron P. Finite permutation groups and finite simple groups metron college. Oxford OXI. 1981;4JD:1-22.

Wisbauer. Robert, foundations of module and ring theory a handbook for study and research. Gordon and Breach Publishers; 1991.

Isaacs IM. Character theory of finite groups. Academic Press, Inc.; 1976.

Peeters R. Uniqueness of strongly regular graphs having minimal p-rank. Linear Algebra Appl. 1995;226228:9-31.

Rodrigues BG. Codes of designs and graphs from finite simple groups. PhD Thesis, University of Natal, Pietermaritzburg; 2002.

Seiple D. Investigation of binary self-dual codes invariant under simple groups. Master’s Thesis, The University of Arizona; 2009.

Key JD, Moori J, Rodrigues BG. On some designs and codes from primitive rep- resentations of some finite simple groups, J. Combin. Math. Combin. Comput. 2003;45:3-19.

Moori J, Rodrigues BG. Some designs and codes invariant under the simple group Co2, J. Algebra. 2007;316:649661.

Grassl Markus. Bounds on the minimum distance of linear codes and quantum codes. Available: http://www.codetables.de (Accessed on 2019-05-06)