Howell designs with sub-designs

Abstract

A Howell design of side s and order 2n, or more briefly an H(s, 2n), is an s × s array in which each cell is either empty or contains an unordered pair of elements from some (2 n)-set V such that (1) every element 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) that n⩽ s⩽2 n − 1. The two boundary cases are well known designs: an H(2n − 1, 2n) is a Room square of side 2 n − 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 interested in the existence of Howell designs which contain as a subarray another Howell design. The existence of Room squares with Room square sub-designs and a pair of mutually orthogonal Latin squares with Latin square sub-designs has been investigated. In this paper, we consider the general problem of constructing H(s, 2n) which contain as sub-designs H(t, 2m). We describe some bounds on the parameters and several constructions for the general case, then we concentrate on determining the spectrum for Howell designs where t= m or t=2 m−1.