On the existence of three dimensional Room frames and Howell cubes

A Howell design of side s and order 2n+2, or more briefly an H(s, 2n+2) isan s× s array in which each cell is either empty or contains an unordered pairof elements from some 2n+2 set V such that (1) every element of V occurs inprecisely one cell of each row and each column, and (2) every unordered pairof elements from V is in at most one cell of the array. It follows immediatelyfrom the definition of an H(s, 2n+2) that n+1 ≤ s ≤ 2n+1. A d-dimensionalHowell 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 some2n+ 2 set V , and (2) each two-dimensional projection is an H(s, 2n+ 2). Thetwo boundary cases are well known designs: an Hd(2n + 1, 2n + 2) is a Roomd-cube of side 2n+ 1 and the existence of d mutually orthogonal latin squaresof 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 completelydetermine 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 ≥ 5with only a few small possible exceptions for v.