| Many open questions in secret sharing ask about the information rate of a particular access structure or class of access structures. Here we improve the best known upper bounds on the information rates of the access structures induced by the Vamos matroid from 10/11 to 8/9 for V1 and from 9/10 to 17/19 for V6. The method we introduce to obtain the bound for V6 can be generalized and applied to all other known 4-variable non-Shannon information inequalities. We also find the exact information rates for the infinitely many minor-minimal, non-matroid-related access structures whose rates were not previously known. On the topic of information inequalities, we show that the Ingleton inequality holds under certain independence assumptions and use this formulation to get a new proof of the Zhang-Yeung non-Shannon inequality. |
