ABSTRACT
X = {a, b, c} or {a, b, c, d}. It is constructed 3-or 4-diagonally cyclic modulo 10. The holes are based on {0, 5}, {1, 6}, {2, 7}, {3, 8}, {4, 9}, and X.
3 8 7 1 9 c b a 2 4 6 a 4 9 8 2 0 c b 3 5 7 b a 5 0 9 3 1 c 4 6 8 c b a 6 1 0 4 2 5 7 9 3 c b a 7 2 1 5 6 8 0 4 c b a 8 3 2 6 7 9 1
7 5 c b a 9 4 3 8 0 2 4 8 6 c b a 0 5 9 1 3 6 5 9 7 c b a 1 0 2 4 2 7 6 0 8 c b a 1 3 5 1 2 3 4 5 6 7 8 9 0 8 9 0 1 2 3 4 5 6 7 9 0 1 2 3 4 5 6 7 8
7 a b 3 c 8 6 d 2 9 1 4 d 8 a b 4 c 9 7 3 0 2 5 8 d 9 a b 5 c 0 4 1 3 6 1 9 d 0 a b 6 c 5 2 4 7 c 2 0 d 1 a b 7 6 3 5 8 c 3 1 d 2 a b 8 7 4 6 9
9 c 4 2 d 3 a b 8 5 7 0 b 0 c 5 3 d 4 a 9 6 8 1 a b 1 c 6 4 d 5 0 7 9 2 6 a b 2 c 7 5 d 1 8 0 3 4 5 6 7 8 9 0 1 2 3 2 3 4 5 6 7 8 9 0 1 7 8 9 0 1 2 3 4 5 6 3 4 5 6 7 8 9 0 1 2
5.18 Theorem
1. If there exists an HSOLS(anb1), then n ≥ 1 + 2b/a. 2. [2174] For n ≥ 4 and a ≥ 2, an HSOLS(anb1) exists if 0 ≤ b ≤ a(n − 1)/2 with
possible exceptions for n ∈ {6, 14, 18, 22} and b = a(n − 1)/2.
