Improved upper bounds for the information rates of the secret sharing schemes induced by the Vámos matroid

An access structure specifying the qualified sets of a secret sharing scheme must have information rate less than or equal to one. The Vámos matroid induces two non-isomorphic access structures V1V1 and V6V6, which were shown by Martí-Farré and Padró to have information rates of at least 3/4. Beimel, Livne, and Padró showed that the information rates of V1V1 and V6V6 are bounded above by 10/11 and 9/10 respectively. Here we improve those upper bounds to 8/9 for V1V1 and 17/19 for V6V6. We also indicate a general method that allows one to read off an upper bound for the information rate of V6V6 directly from the coefficients of any non-Shannon inequality with certain properties, properties that hold for all 4-variable non-Shannon inequalities known to the author.