Room square patterns

Abstract

Abstract Suppose a Howell design H(s,2n) , H , contains as a subarray an m×m array M which contains a room square of order m and possibly other pairs in the “empty” cells of the RS (m) . Then we say that H contains a Room square pattern of order m , an RSP (m) . (i.e. H contains the non-empty cells of an RS (m) .) If M contains only the pairs of an RS (m) , then H contains an RS (m) as a sub-design. In this paper, we are interested in the existence of Howell designs with RSP sub-designs. This investigation is motivated by the fact that RSP sub-designs occur naturally in several constructions for Howell designs; in addition, it is possible to construct H(n,n+α) with RSP (m) sub-designs for values of n and m where H(n,n+α) with RS (m) sub-designs cannot exist.