Error-correcting codes from 2-representations of the unitary group U(3,3)

Abstract

In this dissertation, we use modular representation theory to find error-correcting codes admitting finite simple group as a primitive permutation group and show that every binary linear code admitting group G as a primitive permutation group is a submodule of the permutation module of the primitive action of the group. If the Schur multiplier of the group G is trivial and P is a permutation module of degree n, then every binary linear code of length n invariant under G is a submodule of P. As an illustrative example, we select the finite simple group G = U(3, 3) which is referred to by other authors as PSU(3, 3) and identify the complete set of linear codes derived from its 2-representations. We will find the maximal subgroups of the simple group G = U(3, 3). After finding the maximal subgroups we find the permutation representation, each permutation representation has a corresponding permutation module which we will find. Our computations are based on MAGMA. We then classify these codes and determine their properties such as the minimum distance, minimum weight and the support and other properties. Then we will discuss whether a certain code has good error-correcting or error-detecting abilities based on their properties. In addition, we use the supports of the codes to construct certain designs that remain invariant under the action of U(3, 3) and establish connections between these designs and the corresponding linear codes.