## 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

|G

where G

An entropic vector is (abelian) group representable if there exist a(n) (abelian) group G and subgroups G

|G

_{1}| |G_{2}| |G_{34}| |G_{123}| |G_{124}| ≥ |G_{12}| |G_{13}| |G_{14}| |G_{23}| |G_{24}|where G

_{i}denotes a subgroup of G and G_{ij}denotes the intersection of G_{i}and G_{j}. 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 S_{5}. 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

_{1},..., G

_{n}such that H(X

_{A})= log[G:G

_{A}] 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

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.## Publications

- Journal
**[J1]**E. Thomas, N. Markin, F. Oggier, On Abelian Group Representability of Finite Groups,*Advances in Mathematics of Communication*, volume 8, no 2, 2014.- Conferences
**[C7]**E. Thomas, F. Oggier, Applications of Quasi-uniform Codes to Storage,*Invited Paper, SPCOM 2014*.**[C6]**M. A. Vázquez-Castro, F. Oggier, Lattice Network Coding over Euclidean Domains,*EUSIPCO 2014*.**[C5]**N. Markin, E. Thomas, F. Oggier, Groups and Information Inequalities in 5 Variables,*Allerton 2013*.**[C4]**F. Oggier, J.-C. Belfiore, Enabling Multiplication in Lattice Codes via Construction A,*ITW 2013*.**[C3]**E. Thomas, F. Oggier, Explicit Constructions of Quasi-Uniform Codes from Groups,*ISIT 2013*.**[C2]**E. Thomas, F. Oggier, A Note on Quasi-Uniform Distributions and Abelian Group Representability,*SPCOM 2012*.-
**[C1]**R. Stancu, F. Oggier, Finite Nilpotent and Metacyclic Groups never violate the Ingleton Inequality,*NetCod 2012*. - Invited Talk
- On Groups, Entropy and Network Codes, Campinas University, Brazil, March 13 2012.