New Designs with Block Size 7

Halving block designs with block size four

B) is said to have the Size Four property if the blocks in B can be partitioned into two isomorphic sets. In this paper we construct block with four with the property, thus the exception of 6 A. Rosa [4]).

Embedding handcuffed designs with block size

Let (W, C) be an m-cycle system of order n and let Ω ⊂ W , |Ω| = v < n. We say that a handcuffed design (Ω,P) of order v and block size s (2 ≤ s ≤ m−1) is contained in (W, C) if for every p ∈ P there is an m-cycle c = (a1, a2, . . . , am) ∈ C such that: (1) p = [ak, ak+1, . . . , ak+s−1] for some k ∈ {1, 2, . . . ,m} (i.e. the (s− 1)-path p occurs in the m-cycle c); and (2) ak−1, ak+s ∈ Ω. Note...

On directed designs with block size five

In a t-(v, k, λ) directed design the blocks are ordered k-tuples and every ordered t-tuple of distinct points occurs in exactly λ blocks (as a subsequence). We study t-(v, 5, 1) directed designs with t = 3 and t = 4. In particular, we construct the first known examples, and an infinite class, of 3-(v, 5, 1) directed designs, and the first known infinite class of 4-(v, 5, 1) directed designs.

Perfect Mendelsohn designs with block size six

Balanced incomplete block designs with block size~9: Part III

The necessary conditions for the existence of a balanced incomplete block design on v points, with index λ and block size k, are that: λ(v − 1) ≡ 0 mod (k − 1) λv(v − 1) ≡ 0 mod k(k − 1) Earlier work has studied k = 9 with λ ∈ {1, 2, 3, 4, 6, 8, 9, 12}. In this article we show that the necessary conditions are sufficient for λ = 9 and every other λ not previously studied.