Some Howell Designs of prime side II

Abstract

In part I of this paper we showed that for any pair ( m, r) of positive integers, r odd, there is a positive integer N 1( m, r) such that if p is a prime, p = 2 m rs+1> N 1( m>, r), r, s odd positive integers, then Howell Designs of all types H ∗(p, 2n) , p+1⩽2 n⩽2 p−2 s exist. We now verify that under the same hypotheses (except that ( m>, r) = (1, 1) is not allowed), there is a positive integer N 2( m>, r) such that if p = 2 m rs+1> N 2( m, r), then Howell Designs of all types H ∗(p, 2n) . 2 p−2 s⩽2 n⩽2 p−6 exist. Since designs of type H ∗(p>, 2p−4) and H ∗(p, 2p) are known to exist for p⩾7, it follows that if p = 2 m rs+1> N( m, r) = max{ N 1( m, r). N 2( m, r)}, then Howell Designs of all types H ∗(p, 2n) , except possibly type H ∗(p, 2p−2) , exist. When this is the case, we say that almost all H ∗(p, 2n) exist. As in part I, the method of construction appears to be much better than the general bounds obtained. We are able to show that N(2, 1) = 5. Thus, if 5< p = 4 s+1, s odd, then almost all H ∗(p, 2n) exist. Evidence is presented to show that N(1, 3) is almost certainly 1. In particular, if p = 6 s+1<1000, s odd, then almost all H ∗(p, 2n) exist.