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 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 S5 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 ≥ 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. Index Terms Finite groups, entropy vectors, Ingleton inequality, network coding, network information theory. Portions of this work were presented at the Forty-Seventh Annual Allerton Conference on Communication, Control, and Computing, 2009 and the 2010 IEEE International Symposium on Information Theory. The authors are with the Department of Electrical Engineering, California Institute of Technology, Pasadena, CA 91125 USA (email: [email protected], [email protected], [email protected]). This work was supported in part by the National Science Foundation under grants CCF-0729203, CNS-0932428 and CCF1018927, by the Office of Naval Research under the MURI grant N00014-08-1-0747, and by Caltech’s Lee Center for Advanced Networking. 2