## Introduction

This project looks at two aspects of network coding: the connections between group theory and information theory via the study of quasi-uniform distributions, and the constructions of lattices to enable physical layer network coding.

## The Ingleton Inequality for Groups

The Ingleton Inequality for groups states that given a finite group G, then
|G1| |G2| |G34| |G123| |G124| ≥ |G12| |G13| |G14| |G23| |G24|
where Gi denotes a subgroup of G and Gij denotes the intersection of Gi and Gj. The question is to find groups G which contain at least one instance of 4 subgroups for which the inequality does not hold. It was shown by Chan than Ingleton always holds for abelian groups, and by Mao and Hassibi through computer search that the group with smallest order which violates Ingleton is the symmetric group S5. We showed in [C1] that nilpotent groups and metacyclic groups never violate Ingleton. In [C5] we address the question of Ingleton Inequalities for 5 subgroups.

An entropic vector is (abelian) group representable if there exist a(n) (abelian) group G and subgroups G1,..., Gn such that H(XA)= log[G:GA] for all subsets A of indices in {1...n}. In [C2], we give examples of non-abelian groups whose corresponding entropic vectors could have been obtained from abelian groups, and address the question of determining which non-abelian groups give or not richer entropic vectors than abelian groups. We discuss in particular dihedral groups, which do give richer entropic vectors only when their order is not a power of 2, and not otherwise. In [J1], we completely characterize dihedral, quasi-dihedral and dicyclic groups with respect to their abelian representability, as well as the case when n=2, for which we show a group is abelian representable if and only if it is nilpotent.

## Quasi-Uniform Codes from Groups

We study explicit constructions of quasi-uniform codes from groups and their basic properties in [C3]. Quasi-uniform codes allow codewords to live in different alphabets. We study quasi-uniform codes from this point of view in [C7], and consider its potential application to design storage codes.

## Physical Layer Network Coding

We study lattices and their applications to physical layer network coding. In [C4], we showed how multiplication can be obtained via Construction A, assuming the lattice considered is obtained from an algebraic number field.

In [C6], we look at a different aspect, and build alphabets on quadratic imaginary Euclidean domains.