Abstract

Let v, k and λ be positive integers. A (v, k, λ)-Mendelsohn design (briefly (v, k, λ)-MD) is a pair (X, ) where X is a v-set (of points) and is a collection of cyclically ordered k-subsets of X (called blocks) such that every ordered pair of points of X are consecutive in exactly λ blocks of . A set of k distinct elements {a₁, a₂, …, a_k} is said to be cyclically ordered by a₁<a₂<…<a_k<a₁ and the pair a_i, a_i+t are said to be t-part in a cyclic k-tuple (a₁, a₂,…,a_k) where i + t is taken modulo k. If for all t = 1, 2,…,k − 1, every ordered pair of points of X are t-apart in exactly λ blocks of , then the (v, k, λ)-MD is called perfect and is denoted briefly by (v, k, λ)-PMD. A necessary condition for the existence of a (v, k, λ)-PMD is λv(v−1)≡0 (mod k). In this paper, we shall be concerned mostly with the case k = 7 and λ even. It will be shown that the necessary condition for the existence of a v, 7, λ)-PMD, namely λv(v−1)≡0 (mod 7), is also sufficient for all even λ 16, with at most 29 possible exceptions for the pair (v, λ) where λ is even and λ < 16. In the process, we shall also establish that the necessary condition v≡0 or 1 (mod 7) for the existence of a (v, 7, 1)-PMD is also sufficient for all v 421 with at most 40 possible exceptions below this value, which improves the earlier results.

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