Construction of Balanced Incomplete Block Designs with Binary and Ternary Variance

Abstract

This chapter proposes a new method for constructing binary and ternary variance balanced incomplete block designs based on the Kronecker product of two Hadamard matrices and unreduced symmetrical Balanced Incomplete Block Design.  Method 1 discusses the construction of binary variance balanced block designs using the Kronecker product of two Hadamard matrices, and method 2 discusses the constructions of ternary variance balanced block design using the unreduced symmetrical, balanced incomplete block design with the parameters \(\mathit{v}\) = \(\mathit{b}\), \(\mathit{r}\) = \(\mathit{k}\) = \(\mathit{v}\) - 1, \(\lambda\) = \(\mathit{v}\) - 2 . The optimality of the constructed binary and ternary designs has been studied with numerical examples, and the constructed designs were found to be universally optimal. The built design is significant because it lowers the cost and duration of the experimentation process, increases the accuracy and efficiency of the experimental results, and enables a set of experimental treatments requiring fewer observations to be compared to a full block design.