On the existence of perfect Mendelsohn designs with k = 7 and λ even

Abstract

Let v, k and λ be positive integers. A ( v, k, λ)- Mendelsohn design (briefly ( v, k, λ)-MD) is a pair ( X,B) where X is a v-set (of points) andBis 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 B. A set of k distinct elements { a 1, a 2, ..., a k } is said to be cyclically ordered by a 1< a 2<...< a k < a 1 and the pair a i , a i+ t are said to be t-part in a cyclic k-tuple ( a 1, a 2,..., 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 B, 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 λ v( 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 λ v( 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.