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

An access structure specifying the qualified sets of a secret sharing scheme must have information rate less than or equal to one. The Vلmos matroid induces two non-isomorphic access structures V 1 and V 6 , which were shown by Martي-Farré and Padrَ to have information rates of at least 3/4. Beimel, Livne, and Padrَ showed that the information rates of V 1 and V 6 are bounded above by 10/11 and 9/10 respectively. Here we improve those upper bounds to 8/9 for V 1 and 17/19 for V 6 . We also indicate a general method that allows one to read off an upper bound for the information rate of V 6 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.