Violating the Ingleton Inequality with Finite Groups

TL;DR

This paper investigates finite groups that produce entropy vectors violating the Ingleton inequality, revealing that nonabelian groups like S_5 and PGL(2,p) can surpass linear network code limitations.

Contribution

It identifies specific nonabelian finite groups, including S_5 and PGL(2,p), that generate entropy vectors violating the Ingleton inequality, expanding the understanding of network coding capabilities.

Findings

S_5 is the smallest group violating Ingleton inequality.

The group PGL(2,p) with p > 3 also violates Ingleton.

Nonabelian groups can produce more powerful network codes.

Abstract

It is well known that there is a one-to-one correspondence between the entropy vector of a collection of n random variables and a certain group-characterizable vector obtained from a finite group and n of its subgroups. However, if one restricts attention to abelian groups then not all entropy vectors can be obtained. This is an explanation for the fact shown by Dougherty et al that linear network codes cannot achieve capacity in general network coding problems (since linear network codes form an abelian group). All abelian group-characterizable vectors, and by fiat all entropy vectors generated by linear network codes, satisfy a linear inequality called the Ingleton inequality. In this paper, we study the problem of finding nonabelian finite groups that yield characterizable vectors which violate the Ingleton inequality. Using a refined computer search, we find the symmetric group S_5 to be the smallest group that violates the Ingleton inequality. Careful study of the structure of this group, and its subgroups, reveals that it belongs to the Ingleton-violating family PGL(2,p) with primes p > 3, i.e., the projective group of 2 by 2 nonsingular matrices with entries in F_p. This family of groups is therefore a good candidate for constructing network codes more powerful than linear network codes.