THE EXISTENCE OF SKEW HOWELL DESIGNS OF SIDE 2N AND ORDER 2N+2

A Howell design of side s and order 2n, or more briefly an H(s, 2n), 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 ≤ 2n -1. The existence question for H(2n, 2n + 2) was settled in 1977 by Schellenberg, van Rees, and Vanstone and the existence question for complementary H(2n, 2n + 2) was settled in 1985 by Lamken and Vanstone: for n a positive integer, n ≥ 2, there exists a complementary H(2n, 2n + 2). The existence of *complementary H(2n, 2n + 2) was also established with a finite number of possible exceptions. In this paper, we prove the existence of skew H(2n, 2n + 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(2n, 2n + 2)s to prove the existence of *complementary H(2n, 2n + 2)s for all positive integers n, n ≥ 2. © 1990.