Some improved bounds on the information rate of perfect secret sharing schemes

Informally, a secret sharing scheme is a method of sharing a secret key K among a finite set of participants, in such a way that certain specified subsets of participants can compute a key. Suppose that P is the set of participants. Denote by Γ the set of subsets of participants which we desire to be able to determine the key; hence Γ ⊑ 2P. Γ is called the access structure of the secret sharing scheme. It seems reasonable to require that Γ be monotone, i.e. $$ ifB \in \Gamma andB \subseteq C \subseteq P,thenC \in \Gamma . $$ For any Γ0 ⊑ 2P, define the cioswe of Γ0 to be $$ cl\left( {\Gamma _0 } \right) = \left\{ {C:B \in \Gamma andB \subseteq C \subseteq P} \right\}. $$ Note that the closure of any set of subsets is monotone.

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