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 Vamos matroid induces two non-isomorphic access structures V"1 and V"6, which were shown by Marti-Farre and Padro to have information rates of at least 3/4. Beimel, Livne, and Padro 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.

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