On Partially Balanced Linked 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...

Splitting balanced incomplete block designs

On construction of partially balanced n-ary designs

A new method of constructing a series of partially balanced ternary (PBT) designs is presented. In the method, we have added the corresponding rows of incidence matrices of a BIB design and a PBIB design, both obtained from single initial block with at least one element in common between them. The BIB and PBIB designs above were obtained by method of differences. We have also constructed PBT de...

Linked Balanced Designs are Symmetric BIBD’s

Abstract Let A be a reduced incidence relation between n lines and m points. Suppose that (a) Through each two points there pass λ lines (b) Each two lines intersect in μ points. If λ μ = = 1 assume, moreover, that there are four points no three of which are on one line. Then n m = , λ μ = , and there is a number r such that all lines have r points and through each point there pass r lines. The...

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...