Ideal perfect threshold schemes and MDS codes

Secret sharing schemes (SSS) made their appearance in the form of threshold (n,τ)-schemes in 1979. R. McEliece and D. Sarwate pointed out a relationship between threshold schemes and MDS-codes in 1981. In 1983 Karnin, Greene and Hellman gave an information-theoretic approach to SSS and proved some upper and lower bounds on the number of participants in an ideal perfect threshold SSS. The proof is based, in fact, on the observation that each ideal perfect threshold SSS determines a unique MDS code, and vice versa, when the secret and shadows belong to the same finite field. Brickell and Davenport (see Journal of Cryptology, vol.4, p.123, 1991) considered combinatorial ideal perfect SSS for the general access structure and established the relationship between such schemes and mastroids. From their results the equivalence of combinatorial ideal perfect threshold SSS and MDS codes (i.e. orthogonal arrays OA₁(τ,n+l,q)) follows almost immediately. We give an independent, self-contained proof (following the ideas of Karnin et al.) for the (formally) more general information-theoretic definition of ideal SSS