On the existence of three dimensional Room frames and Howell cubes

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