2-MODULAR REPRESENTATIONS OF UNITARY GROUP U3(4) AS LINEAR CODES

Abstract

A monumental achievement in group theory was done with the announcement of the completion of classification of simple finite groups in 2004. The proof of this work which was termed, a theorem, consists of tens of thousands of pages in several hundred journal articles written by about 100 authors, published between 1995 and 2004. Such voluminous work cannot be understood by any single person. Attempts to simplify the proof has already been embarked on. It is thought that a knowledge of internal structures associated with the groups and more so representation theoretic methods, could go along way to help simplify the proof. This has sparked research of combinatorial objects like codes obtained from groups and their interplay. This thesis is a study of linear binary codes obtained from primitive permutation representations of the simple finite classical group U3(4). Using the established magma databases and the Meataxe software, we consider for each primitive representation over F2, the permutation module obtained from the action of the group on the cosets of its maximal subgroups and the subsequent maximal submodules. Each submodule constitutes a binary code invariant under the group. In this thesis we study linear binary codes, designs and graphs obtained from the group U3(4). Using modular theoretic methods , we construct and enumerate all linear binary codes and designs from primitive permutation representations of degrees 208 and 416 and classify most of the codes. Furthermore, we determine their properties and establish the interplay between these codes and other combinatorial objects like designs and graphs. In the process, we have uncovered the lattice structure of the submodules. We have also determined the full automorphism groups of the codes and designs. Codes are applied in many areas particularly in error correction, storage and transmission of data. The properties of a code determines its usage. We found some codes with good parameters. We found some self-orthogonal, doubly even codes, irreducible and decomposable codes.