Derived Reduced Balanced Incomplete Block Design

Abstract

Construction of Balanced Incomplete Block Designs (BIBD) is a combination problem that involves the arrangement of \(\mathit{v}\) treatments into b blocks each of size \(\mathit{k}\) such that each treatment is replicated exactly \(\mathit{r}\) times in the design and a pair of treatments occur together in \(\lambda\) blocks. Several methods of constructing BIBDs exist. However, these methods still cannot be used to design all BIBDs. Therefore, several BIBDs are still unknown because a definite construction method for all BIBDs is still unknown. The study aimed to develop a new construction method that could aid in constructing more BIBDs. The study derived a new class of BIBD from un-reduced BIBD with parameters \(\mathit{v}\) and \(\mathit{k}\) such that \(\mathit{k} \ge\) 3 through selection of all blocks within the un-reduced BIBD that contains a particular treatment \(\mathit{i}\) then in the selected blocks treatment delete treatment \(\mathit{i}\) and retain all the other treatments. The resulting BIBD was Derived Reduced BIBD with parameters \(v^*=v-1, b^*=\left(\begin{array}{c}v-1 \\ k-1\end{array}\right), k^*=k-1, r^*=\left(\begin{array}{c}v-2 \\ k-2\end{array}\right), \lambda=\left(\begin{array}{c}v-3 \\ k-3\end{array}\right)\). In conclusion, the construction method was simple and could be used to construct several BIBDs, which could assist in solving the problem of BIBD, whose existence is still unknown.