On the existence of three dimensional Room frames and Howell cubes

Abstract

A Howell design of side s and order 2n+2, or more briefly an H(s,2n+2), is an s×s array in which each cell is either empty or contains an unordered pair of elements from some 2n+2 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+2) that n+1≤s≤2n+1. A d-dimensional Howell design Hd(s,2n+2) is a d-dimensional array of side s such that (1) every cell is either empty or contains an unordered pair of elements from some 2n+2 set V, and (2) each two-dimensional projection is an H(s,2n+2). The two boundary cases are well known designs: an Hd(2n+1,2n+2) is a Room d-cube of side 2n+1 and the existence of d mutually orthogonal latin squares of order n+1 implies the existence of an Hd(n+1,2n+2). In this paper, we investigate the existence of Howell cubes, H3(s,2n+2). We completely determine the spectrum for H3(2n,2n+α) where α∈{2,4,6,8}. In addition, we establish the existence of 3-dimensional Room frames of type 2v for all v≥5 with only a few small possible exceptions for v.