Derived Reduced Balanced Incomplete Block Design

Abstract

Construction of Balanced Incomplete Block Designs (BIBD) is a combination problem that involves the arrangement of 𝑣 treatments into 𝑏 blocks each of size 𝑘 such that each treatment is replicated exactly 𝑟 times in the design and a pair of treatments occur together in 𝜆 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 𝑣 and 𝑘 such that 𝑘 ≥ 3 through selection of all blocks within the un-reduced BIBD that contains a particular treatment 𝑖 then in the selected blocks treatment delete treatment 𝑖 and retain all the other treatments. The resulting BIBD was Derived Reduced BIBD with parameters 𝑣 ∗ = 𝑣 − 1, 𝑏 ∗ = ( 𝑣−1 𝑘−1 ), 𝑘 ∗ = 𝑘 − 1, 𝑟 ∗ = ( 𝑣−2 𝑘−2 ), 𝜆 = ( 𝑣−3 𝑘−3 ) . 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