Residual Reduced Balanced Incomplete Block Design

Abstract

The construction of Balanced Incomplete Block Designs is a combination problem that involves the arrangement of <i>v</i> treatments into <i>b</i> blocks each of size <i>k</i> such that each treatment is replicated exactly <i>r</i> times in the design and a pair of treatments occur together in λ blocks. Researchers have devised a number of methods that can be used in constructing BIBDs, using geometry, difference sets, existing BIBD designs, computers and mathematical algorithms, and Latin squares. However, the existing constructing methods still cannot be used to construct all the BIBDs. This has left the existence of some BIBDs to still be unknown as some of them still cannot be constructed using the existing construction methods. The study aimed to derive a new construction method that uses the un-reduced BIBD to construct a new class of BIBD known as Residual Reduced BIBD. The study used the un-reduced BIBD with parameters (<i>v, k</i>) to construct the new class of BIBD. Consider an un-reduced BIBD with parameters <i>v</i> and <i>k</i> such that k≥3 the Residual Reduced BIBD was derived from the un-reduced design selection of blocks of the un-reduced BIBD that contain a particular treatment <i>i</i>. Then in the selected blocks if treatments <i>i</i> deleted and the rest of the treatments are left, then this forms a BIBD known as Residual Reduced BIBD. Residual Reduced BIBD formed has the parameters<i> v</i>^* = <i>v</i> -1, <i>b</i>^* = ((<i>v</i> - 1)!(<i>v</i> - <i>k</i>))/(<i>k</i>!(<i>v</i> - <i>k</i>)!), <i>k</i>^* = <i>k</i>, <i>r</i>^* = ((<i>v</i> - 2)!(<i>v</i> - <i>k</i>))/((<i>k</i> - 1)!(<i>v</i> - <i>k</i>)!), <i>λ</i>^* = ((<i>v</i> - 3)!(<i>v</i> - <i>k</i>))/((<i>k</i> - 2)!(<i>v</i> - <i>k</i>)!). In conclusion, the study was able to show that a new class of BIBD could be constructed from the un-reduced BIBD. This means that some other BIBDs still can be derived from this universal set using appropriate procedures.