The existence of skew Howell designs of side 2n and order \(2n+2\)
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. From the definition of an \(H(s,2n+2)\), \(n+1\leq s\leq 2n+1\). Howell designs have been studied extensively and the existence of these designs was determined completely in \textit{B. A. Anderson, P. J. Schellenberg} and \textit{D. R. Stinson} [J. Comb. Theory, Ser. A 36, 23-55 (1984; Zbl 0527.05013)] and \textit{D. R. Stinson} [J. Comb. Theory, Ser. A 32, 53-65 (1982; Zbl 0472.05012)]. In this paper, the authors prove that the existence of skew \(H(2n,2n+2)s\) for n a positive integer, \(n\geq 2\), \(n\neq 3\) with the possible exceptions of \(n=5\) and \(n=9\), by using an intransitive starter-adder construction. The authors also improve the existence result for \({}^*\) complementary \(H(2n,2n+2)s\) to prove the existence of \({}^*\) complementary \(H(2n,2n+2)s\) for all positive integers \(n\geq 2\).
scientific article; zbMATH DE number 4143985
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