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On the existence of three dimensional room frames and Howell cubes. (English) Zbl 1279.05011

Summary: A Howell design of side \(s\) and order \(2n+2\), or more briefly an \(H(s,2n+2)\), is an \(s\times s\) 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 \(n+1\leq s\leq 2n+1\). 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 \(H_d(2n+1,2n+2)\) 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 \(H_d(n+1,2n+2)\).
In this paper, we investigate the existence of Howell cubes, \(H_3(s,2n+2)\). We completely determine the spectrum for \(H_3(2n,2n+\alpha )\) where \(\alpha\in\{2,4,6,8\}\). In addition, we establish the existence of 3-dimensional Room frames of type \(2^v\) for all \(v\geq 5\) with only a few small possible exceptions for \(v\).

MSC:

05B30 Other designs, configurations
05B05 Combinatorial aspects of block designs
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