Large violations of the Ingleton inequality

In network information theory, non-Shannon-type inequalities arise in determining capacity/entropy regions where there are more than three random variables. The most famous such inequality is the Ingleton inequality, which is satisfied by linear network codes. To produce better network codes raises the question of finding points in the region that violate the Ingleton inequality and this problem can be translated into group theory. Abelian and small groups do not produce violations, and Mao, Thill, and Hassibi computed that the smallest Ingleton-violating group is the symmetric group on five letters, of order 120. They then generalized this example to other matrix groups. In each of their cases, the Ingleton ratio (which is less than or equal to 1 if and only if the inequality holds) is smaller than 4/3. We go beyond their work to show that examples of Ingleton-violating groups abound, construct explicit examples with arbitrarily large Ingleton ratio, give a systematic approach to finding good examples and a much simplified formula for the Ingleton ratio, and give supporting evidence for the Four-Atom Conjecture of Dougherty, Freiling, and Zeger, which would bound the ratio in terms of the group order.