HOWELL DESIGNS WITH SUB-DESIGNS

A Howell design of side s and order 2n, or more briefly an H(s, 2n), i s an s x s array in which each cell is either empty or contains an uno rdered pair of elements from some (2n)-set V such that (1) every eleme nt of V occurs in precisely one cell of each row and each column, and (2) every unordered pair of elements from V is in at most one cell of the array. It follows immediately from the definition of an H(s, 2n) t hat n less-than-or-equal-to s less-than-or-equal-to 2n-1. The two boun dary cases are well known designs: an H(2n - 1, 2n) is a Room square o f side 2n - 1 and the existence of a pair of mutually orthogonal Latin squares of order n implies the existence of an H(n, 2n). We are inter ested in the existence of Howell designs which contain as a subarray a nother Howell design. The existence of Room squares with Room square s ub-designs and a pair of mutually orthogonal Latin squares with Latin square sub-designs has been investigated. In this paper, we consider t he general problem of constructing H(s, 2n) which contain as sub-desig ns H(t, 2m). We describe some bounds on the parameters and several con structions for the general case, then we concentrate on determining th e spectrum for Howell designs where t = m or t = 2m - 1. (C) 1994 Acad emic Press, Inc.