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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. *Corresponding author: E-mail: mojiema@mmust.ac.ke, michaelojiema@yahoo.com; Kariuki et al.; ARJOM, 15(4): 1-17, 2019; Article no.ARJOM.52223 We further obtain the lattice structures of the sub-modules and compare these ternary codes with the binary codes constructed from the same group.

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