2-modular codes admitting the simple group L3(4) as an automorphism group

Abstract

Given a permutation group G on a finite set Ω. and a field F it is often of considerable interest to know the structure of the permutation module Ω (that is, the vector space over with basis Ω considered as an G module). The G-invariant submodules of Ω can be regarded as linear codes in Ω, and one may therefore ask for the weight distribution of these codes. We construct and enumerate all non-trivial binary linear codes from the 2-modular primitive representations of the simple group L₃(4), using a chain of maximal submodules of a permutation module induced by the action of L₃(4) on objects such as the lines, hyperovals, Baer subplanes and unitals of PG₂(4). Several codes with intersting properties are obtained, among these optimal and self-orthogonal codes invariant under L₃(4). We establish results on non-existence of L₃(4)-invariant self-dual codes of lengths 56, 120 and 280 respectively, and moreover that L₃(4) is not realizable as the automorphism group of a binary linear code.