Construction of Symmetric Balanced Incomplete Block Designs Using Mutually Orthogonal Latin Squares and Galois Field GF(4) Methods.

Abstract

t: This study presents a novel approach for the construction and verification of a Symmetric Balanced Incomplete Block Design (SBIBD) using Mutually Orthogonal Latin Squares (MOLS) derived from Galois Field GF(4). Three mutually orthogonal Latin squares of order 4 were generated using finite field transformations and modular arithmetic principles, followed by column and row permutation techniques to preserve orthogonality and balance. The constructed design produced an SBIBD satisfying the standard combinatorial conditions of v = b = 16, constant block size k = r = 4,  and pairwise occurrence parameter λ=1\lambda =1λ=1, thereby ensuring uniform treatment allocation and balanced replication. The orthogonality verification confirmed that all ordered treatment pairs occurred exactly once across the superimposed structures, validating the mathematical consistency of the design. The proposed method reduced the complexity associated with manual BIBD construction while maintaining symmetry, balance, and structural flexibility. The results demonstrate that the GF(4)-based MOLS framework provides a statistically robust and computationally efficient approach for constructing SBIBDs applicable to agricultural experiments, industrial process optimization, and other combinatorial experimental designs. The study, therefore contributes to the advancement of experimental design through the integration of combinatorial mathematics and statistical design theory.