On Ingleton-Violating Finite Groups

Given <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> discrete random variables, its entropy vector is the <inline-formula> <tex-math notation="LaTeX">$2^{n}-1$ </tex-math></inline-formula>-dimensional vector obtained from the joint entropies of all non-empty subsets of the random variables. It is well known that there is a close relation between such an entropy vector and a certain group-characterizable vector obtained from a finite group and <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> of its subgroups; indeed, roughly speaking, knowing the region of all such group-characterizable vectors is equivalent to knowing the region of all entropy vectors. 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 <italic>et al.</italic> that linear network codes cannot achieve capacity in general network coding problems (since linear network codes come from abelian groups). All abelian group-characterizable vectors, and by fiat all entropy vectors generated by linear network codes, satisfy a linear inequality called the Ingleton inequality. General entropy vectors, however, do not necessarily have this property. 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 <inline-formula> <tex-math notation="LaTeX">$S_{5}$ </tex-math></inline-formula> 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 <inline-formula> <tex-math notation="LaTeX">$PGL(2,q)$ </tex-math></inline-formula> with a prime power <inline-formula> <tex-math notation="LaTeX">$q \geq 5$ </tex-math></inline-formula>, i.e., the projective group of <inline-formula> <tex-math notation="LaTeX">$2\times 2$ </tex-math></inline-formula> nonsingular matrices with entries in <inline-formula> <tex-math notation="LaTeX">$ \mathbb {F}_{q}$ </tex-math></inline-formula>. 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 <inline-formula> <tex-math notation="LaTeX">$PGL(n,q)$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$GL(n,q)$ </tex-math></inline-formula>. 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.

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