Inequalities for Shannon entropy and Kolmogorov complexity
Journal of Computer and System Sciences - Eleventh annual conference on computational learning theory&slash;Twelfth Annual IEEE conference on computational complexity
Conditional Independences among Four Random Variables III: Final Conclusion
Combinatorics, Probability and Computing
On characterization of entropy function via information inequalities
IEEE Transactions on Information Theory
On a relation between information inequalities and group theory
IEEE Transactions on Information Theory
Insufficiency of linear coding in network information flow
IEEE Transactions on Information Theory
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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 non-abelian 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.