TITLE:
Construction of Non-Symmetric Balanced Incomplete Block Design through Combination of Symmetric Disjoint Balanced Incomplete Block Designs
AUTHORS:
Troon John Benedict, Onyango Fredrick, Karanjah Anthony, Njuguna Edward
KEYWORDS:
Disjoint Symmetric BIBD, Un-Reduced BIBD, Combination, Symmetric BIBD, Non-Symmetric BIBD, Non-Existence of BIBD
JOURNAL NAME:
Open Journal of Statistics,
Vol.13 No.6,
December
6,
2023
ABSTRACT: 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.