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

作　　者：Troon John Benedict Onyango Fredrick Karanjah Anthony Njuguna Edward Troon John Benedict;Onyango Fredrick;Karanjah Anthony;Njuguna Edward(Department of Economics, Maasai Mara University, Narok, Kenya;Department of Mathematics and Actuarial Science, Maseno University, Luanda, Kenya;Department of Mathematics, Multimedia University, Nairobi, Kenya;Department of Mathematics and Physical Science, Maasai Mara University, Narok, Kenya)

机构地区：[1]Department of Economics, Maasai Mara University, Narok, Kenya [2]Department of Mathematics and Actuarial Science, Maseno University, Luanda, Kenya [3]Department of Mathematics, Multimedia University, Nairobi, Kenya [4]Department of Mathematics and Physical Science, Maasai Mara University, Narok, Kenya

出　　处：《Open Journal of Statistics》2023年第6期789-802,共14页统计学期刊（英文）

摘　　要：The existence of several non-symmetric balanced incomplete block designs (BIBDs) is still unknown. This is because the non-existence property for non-symmetric 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 = b, r = 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, where 2≤ n ≤ r - λ. The study established that the non-existence property of this class of BIBD was that when is not a perfect square then v should be even and when v^*is odd then the equation should not have a solution in integers x, y, z 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.The existence of several non-symmetric balanced incomplete block designs (BIBDs) is still unknown. This is because the non-existence property for non-symmetric 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 = b, r = 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, where 2≤ n ≤ r - λ. The study established that the non-existence property of this class of BIBD was that when is not a perfect square then v should be even and when v^*is odd then the equation should not have a solution in integers x, y, z 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.

关 键 词：Disjoint Symmetric BIBD Un-Reduced BIBD COMBINATION Symmetric BIBD Non-Symmetric BIBD Non-Existence of BIBD 

分 类 号：O17[理学—数学]