The Existence of (ν,6, λ)-Perfect Mendelsohn Designs with λ > 1

Abstract

The basic necessary conditions for the existence of a (v, k, λ)-perfect Mendelsohn design (briefly (v, k, λ)-PMD) are v ≥ k and λv(v − 1) ≡ 0 (mod k). These conditions are known to be sufficient in most cases, but certainly not in all. For k = 3, 4, 5, 7, very extensive investigations of (v, k, λ)-PMDs have resulted in some fairly conclusive results. However, for k = 6 the results have been far from conclusive, especially for the case of λ = 1, which was given some attention in papers by Miao and Zhu [34], and subsequently by Abel et al. [1]. Here we investigate the situation for k = 6 and λ > 1. We find that the necessary conditions, namely v ≥ 6 and λv(v − 1) ≡ 0 (mod 6) are sufficient except for the known impossible cases v = 6 and either λ = 2 or λ odd.Keywords Perfect Mendelsohn design · Incomplete Perfect Mendelsohn design · Holey Perfect Mendelsohn design · GDD · PBD

Resolvable Mendelsohn Designs and Finite Frobenius Groups

“…. p et t ≥ 3 be an integer in prime factorization, and let k be a prime factor of gcd(p e 1 1 − 1, . .…”

“…As mentioned above, a necessary condition for the existence of a (v, k, λ)-MD is that λv(v − 1) ≡ 0 mod k. This condition has been proved to be sufficient for the existence of a (v, k, λ)-PMD when: (i) k = 3, except for the non-existing (6, 3, 1)-PMD [5,23]; (ii) k = 4, except for v = 4 and λ odd, v = 8 and λ = 1 [4, Theorem 1.2]; (iii) k = 5, except for λ = 1, v ∈ {6, 10}, and possibly for λ = 1 and v ∈ {15, 20} [4,Theorem 1.3]. See [1,4,6,25] for more results on PMDs.…”

“…Thus, in view of the proof of Lemma 3.2, the permutation (1 23 52 30)(2 46 51 7) • • • (41 42 12 11) is a 4-regular orthomorphism of Z 53 . In view of the proof of Theorem 4.1, this orthomorphism gives rise to the (53, 4, 1)-RMD B(53) whose basis B 0 (53) consists of(1,23, 52,30),(2, 46, 51,7), . .…”