On the Ingleton-Violating Finite Groups and Group Network Codes

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 [1]. 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 [2] that linear network codes cannot achieve capacity in general network coding problems (since linear network codes form an abelian group). All abelian group-characterizable vector s, and by fiat all entropy vectors generated by linear network codes, satisfy a linear inequality called th e Ingleton inequality. In this paper, we study the problem of finding nonabelian finite groups that yield charac terizable vectors which violate the Ingleton inequality. Using a refined computer search, we find the symme tric group S5 to be the smallest group that violates the Ingleton inequality. Careful study of the stru cture of this group, and its subgroups, reveals that it belongs to the Ingleton-violating family PGL(2,p) with primes p ≥ 5, i.e., the projective group of 2×2 nonsingular matrices with entries in Fp. This family of groups is therefore a good candidate for constructing network codes more powerful than linear network codes.