Some Howell designs of prime side

Abstract

This paper deals with the existence question for types of Howell Designs of odd side. One of the difficulties that must be surmounted before complete knowledge can be obtained is the problem of constructing designs of prime side. Previously, relatively little information has been available about these types of designs. In this paper, we show that for every pair ( m, r) of positive integers, r odd, there is a positive integer P( m, r) such that if p is a prime, p = 2 m rs + 1> P( m, r), r, s odd positive integers, then at least (2 m−1r−1) (2 m−1r) of the types of Howell Designs of side p other than the Room Square type can be constructed. The method of construction appears to be much better than the general bounds that are obtained. We are able to show by ad hoc procedures that P(2,1) = 1, Thus if p = 4 t + 1, t odd, then at least 1 2 of the types of designs of side p other than the Room Square type can be constructed. We offer compelling evidence in favor of the conjecture that P(1,3) = 1 also. In particular, if p = 6 t + 1< 1000, t odd, then at least 3 3 of the types of designs of side p other than the Room Square type can be constructed.