In this note the existence of a (Vj P2j 4, 2) BTD I for P2 = 0, 1 and 2, in which any pair of blocks intersect in at most two elements, is proved for any admissible v. A balanced ternary design is a collection of multi-sets of size k, chosen from a v-set in such a way that each element occurs 0, 1 or 2 times in anyone block, each pair of non-distinct elements, {x, x}, occurs in P2 blocks of the...

Some basics of combinatorial block design are combined with certain constraint satisfaction problems of interest to the satisfiability community. The paper shows how such combinations lead to satisfiability problems, and shows empirically that these are some of the smallest very hard satisfiability problems ever constructed. Partially balanced (0; 1) designs (PB01Ds) are introduced as an extens...

Let H be a subgraph of G. An H-design (V, C) of order v and index μ is embedded into a G-design (X,B) of order v + w and index λ if μ ≤ λ, V ⊆ X and there is an injective mapping f : C → B such that B is subgraph of f (B) for every B ∈ C. For every pair of positive integers v, λ, (except when λ = 3 and v = 30, 34, 42, 46, 54, 58, 66 or λ = 5 and v = 19) we determine the minimum value of w such ...

The design (15,21,7,5,2) is the only BIBD of block size 5 that does not exist. If it did exist, it would provide an exact bicovering, in 21 blocks of size 5, of the pairs from 15 points. However, we show that 22 quintuples are sufficient to provide a bicover of the pairs from 15 points; thus there are only 10 repetitions required in the bicovering.

When the necessary conditions for a BIBD are satisfied, but no BIBD exists, there is no simple answer for the optimal design problem. This paper identifies the E-optimal information matrices for any such irregular BIBD setting when the number of treatments is no larger than 100. Aand D-optimal designs are typically not E-optimal. An E-optimal design for 15 treatments in 21 blocks of size 5 is f...