Quasi-uniform codes and information inequalities using group theory

Abstract

This thesis is dedicated to the study of information inequalities and quasi-uniform codes using group theory. Understanding the region of entropic vectors for dimension n ≥ 4 is an open problem in network information theory. It can be studied using information inequalities and their violations. The connection between entropic vectors and finite groups, known as 'group representability', is a useful tool to compute these violations. In the first part of this thesis we address the problem of extracting 'abelian group representable' vectors out of the whole set of group representable vectors. We prove that certain classes of non-abelian groups are abelian group representable and non-nilpotent groups are not abelian group representable. We then address the question of finding linear inequality violators for n = 5 and obtain the smallest group violators of two linear inequalities. Random variables which are uniformly distributed over their support are known as quasi-uniform. One way of getting quasi-uniform random variables is by using finite groups and subgroups. Codes are constructed in such a way that the associated random variables are quasi-uniform and group theory is used to construct such codes. In the second part of this thesis, we consider the construction of quasi-uniform codes coming from groups and their algebraic properties. We compute some coding parameters and bounds in terms of groups. Finally we propose some applications of quasi-uniform codes especially to distributed storage.