Construction of Non-Symmetric Balanced Incomplete Block Design through Combination of Symmetric Disjoint Balanced Incomplete Block Designs

Abstract

The existence of several non-symmetric balanced incomplete block designs (BIBDs) is still unknown. This is because the non-existence property for nonsymmetric BIBDs is still not known and also the existing construction methods have not been able to construct these designs despite their design parameters satisfying the necessary conditions for the existence of BIBD. The study aimed to bridge this gap by introducing a new class of non-symmetric BIBDs. The proposed class of BIBDs is constructed through the combination of disjoint symmetric BIBDs. The study was able to determine that the total number of disjoint symmetric BIBDs (n) with parameters ( v br k = = , ,λ ) that can be obtained from an un-reduced BIBD with parameters ( v k, ) is given by n r = − λ . When the n symmetric disjoint BIBDs are combined, then a new class of symmetric BIBDs is formed with parameters * v v = , * b nb = , * r nr = , * k k = , * λ λ = n , where 2 ≤ ≤− n r λ . The study established that the non-existence property of this class of BIBD was that when * * r n − λ is not a perfect square then v should be even and when * v is odd then the equation ( ) * * * 1 * 22 2 1 2 v r xy z n n λ λ −    − = +−       should not have a solution in integers xyz , , which are not all simultaneously zero. In conclusion, the study showed that this construction technique can be used to construct some non-symmetric BIBDs. However, one must first construct the disjoint symmetric BIBDs before one can construct the non-symmetric BIBD. Thus, the disjoint symmetric BIBDs must exist first before the non-symmetric BIBDs exist. Keywords Disjoint Symmetric BIBD, Un-Reduced BIBD, Combination, Symmetric BIBD, Non-Symmetric BIBD, Non-Existence of BIBD