A refinement of the Four-Atom Conjecture

In network information theory, Shannon-type inequalities are not enough to describe the entropy regions if there are more than three random variables. The Ingleton inequality is one of the most interesting and useful non-Shannon-type inequalities in four random variables. It is satisfied by linear network codes, but not by all network codes. A measure of how much it fails by is given by the Ingleton score. The Four-Atom Conjecture of R. Dougherty, C Freiling, and K. Zeger states that the Ingleton score cannot exceed 0.089373. Using groups to characterize the entropy region, we propose a two-dimensional extension of the Ingleton score, which yields finer information. In particular, we conjecture constraints on this two-dimensional Ingleton score, i.e. a refinement of the Four-Atom Conjecture. Also, we present two families of examples that between them produce all permissible two-dimensional Ingleton scores.