Geometrical Approach for Construction of Balanced Incomplete Block Design

Abstract

Construction of balanced incomplete block design has been a major concern in the field of combinatorial design. Different techniques have been proposed by several authors to verify the existence and non-existence of balanced incomplete block designs (BIBDs); however, there is still no general method or algorithm to solve this challenge. In this paper, we propose a projective geometry approach of order PG(N, p) to construct balanced incomplete block designs with different N-dimensions. We establish PG(2,2) and PG(3,2) orders to construct two different sets of BIBDs. It is observed that the constructed balanced incomplete block designs (BIBDs) are also symmetry balanced incomplete block designs (SBIBDs), but the reverse is not true in some cases. Hence, this approach is easier and more straightforward for constructing BIBDs and has proven to be effective in determining the existence and non-existence of balanced incomplete block designs.

Keywords: Geometry;Projective Geometry; block design; balanced design; Incomplete balanced design.

2010 Mathematics Subject Classification. 26A25; 26A35