SOME BINARY LINEAR CODES, DESIGNS AND GRAPHS FROM THE ORTHOGONAL GROUPS O- 8(2) AND O+8(2)

THESIS TITLE: SOME BINARY LINEAR CODES, DESIGNS AND GRAPHS FROM THE ORTHOGONAL GROUPS O^– 8(2) AND O+8(2)

STUDENT’S NAME: ELIZABETH NANJALA MASIGA

SUPERVISORS NAMES:

1. LUCY WALINGO CHIKAMAI
2. VINCENT NYONGESA MARANI

ABSTRACT

Simple groups form the building blocks of all other groups and the class of all simple groups is given in a theorem known as classification of finite simple groups. The proof of this theorem, which was a monumental work in group theory involved several researchers and was completed in 1982. It is captured in 500 volumes. In an attempt to better understand and simplify the proof of this theorem, several researchers have embarked on studying the underlying structures of these groups. To this end, a study of the interplay between these groups and other combinatorial structures has recently become one such area of focus. In this work, an action of cosets of the simple groups O-8(2) and O+8(2) on their maximal subgroups is considered. Binary linear codes, designs and graphs are constructed which then become invariant under these groups. Their parameters and properties such as weight and automorphism groups are determined. Two methods have been used, the first one often known as Key-Moori method 1 and the second known as modular theoretic approach. In this second method, all codes invariant under our groups are determined and the lattice structure is also given. Some optimal codes have been found. It is found that no code, design or graph have the automorphism group as its group. We used Magma for our computations. Codes are used in communication and storage because they are used to correct error in these place. Designs are used by statisticians in sampling techniques and in agricultural sciences in the analysis of variance.