The existence of skew Howell designs of side 2 n and order 2 n + 2

Abstract

A Howell design of side s and order 2 n, or more briefly an H( s, 2 n), is an s × s array in which each cell is either empty or else contains an unordered pair of elements from some 2n-set, say X, such that 1. (1) each row and each column is Latin (i.e., every element of X is in precisely one cell of each row and each column) and 2. (2) every unordered pair of elements from X is in at most one cell of the array. Necessary conditions on the parameters s and n are n ⩽ s ⩽ 2 n −1. The existence question for H(2 n, 2 n + 2) was settled in 1977 by Schellenberg, van Rees, and Vanstone and the existence question for complementary H(2 n, 2 n + 2) was settled in 1985 by Lamken and Vanstone: for n a positive integer, n ⩾ 2, there exists a complementary H(2 n, 2 n + 2). The existence of ∗complementary H(2 n, 2 n + 2) was also established with a finite number of possible exceptions. In this paper, we prove the existence of skew H(2 n, 2 n + 2) s for n a positive integer, n ⩾ 2, n ≠ 3 with the possible exceptions of n = 5 and n = 9. We also improve the existence result for ∗complementary H(2 n, 2 n + 2) s to prove the existence of ∗complementary H(2 n, 2 n + 2) s for all positive integers n, n ⩾ 2.