Description
Characterizing the entropy region Γ*[subscript]n is essential to determining network coding capacities. This region can be fully characterized for n = 2 and n = 3 by Shannon-type inequalities. However, there are inequalities involving four random variables which always hold that are not Shannon-type. The region comprised of group-characterizable entropy vectors provides a good approximation of Γ*[subscript]n. It is thus sufficient to study this region. The Ingleton Inequality provides an inner bound on Γ*[subscript]n, but this bound is not tight. Study of group-characterizable entropic vectors that violate the Ingleton Ratio help provide further characterization of the entropy region. We decompose the Ingleton Ratio into the Mixing Ratio and Inclusion-Exclusion Ratio. Two conjectures about these ratios were proposed by Mackenzie. We make progress in understanding these conjectures by testing Ingleton violating examples for consistency with the conjectures, and proving them for specific cases. They remain to be proven in the general case.
