Derivation of Two Associate PBIBD from Necessary Properties of BIBD

Abstract

Abstract: One of the main shortfall of Balanced Incomplete Block Designs is that the design is not available for all parameter sets. Thus, some of the parameter sets that satisfy the necessary conditions for BIBD cannot be constructed as such. This means that the parameter sets for the designs is not possible to be constructed with a single association scheme (λ). Given that the structural difference that exist between BIBD and PBIBD is that the later do not have a single association scheme but rather m association schemes, the study believed that if the BIBD parameter sets could not be constructed as a single associations scheme then maybe if the design is broken down to more than one association scheme in form of a PBIBD then maybe such design might later be constructed using other methods as PBIBDs. The study aimed at determining whether two association scheme PBIBDs could be derived from necessary properties of BIBDs. The main reason for maiking the transformation is that for BIBD some of the designs that satisfy the necessary properties have been determined not to exist meaning that the designs could not be constructed using a single association scheme λ. Therefore, the study felt that if the designs could be broken down into 2 association scheme (λ1 and λ2) then such a design might be constructed as a PBIBD. The study related the necessary properties of BIBD and two association scheme PBIBD. Using the properties, the study created eight sets of linear equations and using the Gauss Jordan Elimination method, the study was able to solve for the eight unknown parameters of the PBIBD association scheme. The study was able in the end to convert a BIBD that satisfy necessary properties of BIBD into a two association scheme PBIBD that satisfy all the necessary properties of PBIBD. Keywords: BIBD, PBIBD, Association Scheme, Necessary Properties, Derivation