On Ingleton-Violating Finite Groups

Abstract

Given n discrete random variables, its entropy vector is the 2 n − 1 dimensional vector obtained from the joint entropies of all non-empty subsets of the random variables. It is well known that there is a one-toone correspondence between such an entropy vector and a certain group-characterizable vector obtained from a finite group and n of its subgroups [3]. This correspondence may be useful for characterizing the space of entropic vectors and for designing network codes. 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[4] 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. It is therefore of interest to identify groups that violate 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 P GL(2, q) with a prime power q ≥ 5, i.e., the projective group of 2 × 2 nonsingular matrices with entries in F q . We further interpret this family of groups, and their subgroups, using the theory of group actions and identify the subgroups as certain stabilizers. We also extend the construction to more general groups such as P GL(n, q) and GL(n, q). The families of groups identified here are therefore good candidates for constructing network codes more powerful than linear network codes, and we discuss some considerations for constructing such group network codes. Index TermsPortions of this work were presented at the

Violations of the Ingleton inequality and revising the four-atom conjecture

“…The first idea is to use group theory to find Ingleton-violating examples, which we describe in Section 3. Unlike the families of violating examples discovered by W. Mao, M. Thill and B. Hassibi [18], whose Ingleton scores approach zero, we find many group violating examples with significantly large Ingleton scores. Among the Ingleton-violating examples we have found, a special phenomenon, factorizability, appears in many of them.…”

“…When inclusion happens between subgroups, i. e., G i ≤ G j for distinct i, j ∈ N 4 , or G i is trivial for some i, the Ingleton inequality holds. Readers are referred to [18] for similar such properties. Note that the Ingleton ratio does not depend on the size of G. One goal is to find large Ingleton ratios in relatively small groups.…”

“…W. Mao, M. Thill and B. Hassibi [18] found examples in non-abelian groups that violate the Ingleton inequality. They discovered that the smallest Ingleton-violating example arises in Sym(5) with Ingleton ratio 16/15 and Ingleton score 0.01348.…”

Entropy Region and Convolution

“…The same surmise appeared later in [14] as Four-atom conjecture, referring to the four possible values of ξ i ξ j ξ k ξ l . The minimization was considered also in [29,3] that report no score below I ij (h p * ). However, the computer experiments discussed in Section 7 found an entropic point that can be transformed to an almost entropic point witnessing failure of Four-atom conjecture.…”

“…Therefore, the infimal score I * is lower bounded by I ij (r ij ) = − Upper bounds on the infimal Ingleton score arise from entropic polymatroids that violate the Ingleton inequality. There are many examples at disposal [40,31,32,46,19,16,18,29,3,43]. The following one has attracted a special attention.…”