Common Treatments Between Blocks of Certain Partially Balanced Incomplete Block Designs

On balanced incomplete-block designs with repeated blocks

Balanced incomplete-block designs (BIBDs) with repeated blocks are studied and constructed. We continue work initiated by van Lint and Ryser in 1972 and pursued by van Lint in 1973. We concentrate on constructing (v, b, r, k, λ)-BIBDs with repeated blocks, especially those with gcd(b, r, λ) = 1 and r ≤ 20. We obtain new bounds for the multiplicity of a block in terms of the parameters of a BIBD...

Splitting balanced incomplete block designs

Remarks on Balanced Incomplete Block Designs

Proof. Assume the hypothesis and the falsity of either conclusion. Construct matrix A of 2x + 2 rows and 4x+4 columns, with entries + 1 and — 1. The first column contains exclusively +1, and the second column — 1. Set up one-to-one correspondences between rows of A and elements of D ; between columns other than the first two of A and blocks of D. Enter +1 if the element is contained in the bloc...

Finding Balanced Incomplete Block Designs with Metaheuristics

This paper deals with the generation of balanced incomplete block designs (BIBD), a hard constrained combinatorial problem with multiple applications. This problem is here formulated as a combinatorial optimization problem (COP) whose solutions are binary matrices. Two different neighborhood structures are defined, based on bit-flipping and position-swapping. These are used within three metaheu...

Beautifully Ordered Balanced Incomplete Block Designs

Beautifully Ordered Balanced Incomplete Block Designs, BOBIBD(v, k, λ, k1, λ1), are defined and the proof is given to show that necessary conditions are sufficient for the existence of BOBIBD with block size k=3 and k1=2 and for k=4 and k1=2 except possibly for eleven exceptions. Existence of BOBIBDs with block size k=4 and k1=3 is demonstrated for all but one infinite family and the non-existe...