Linear codes resulting from finite group actions

In this article, we use group action theory to define some important ternary linear codes. Some of these codes are self-orthogonal having a minimum distance achieving the lower bound in the previous records. Then, we define two new codes sharing the same automorphism group isomorphic to C2 × M11 where M11 is the Sporadic Mathieu group and C2 is a cyclic group of two elements. We also study the natural action of the general linear group GL(k, 2) on the vector space F k 2 to characterize Hamming codes Hk(2) and their automorphism group.