Construction of Symmetric Balanced Squares

Abstract

Symmetric balanced squares for different sizes of array and for different numbers of treatments have been constructed. An algorithm, easily implementable on computers, has been developed for construction of such squares whenever the parameters satisfy the necessary conditions for existence of the square. The method of construction employs $1$-factorizations of a complete graph or near $1$-factorizations of a complete graph, depending on whether the size of the array is even or odd, respectively. For odd sized squares the method provides a solution directly based on the near $1$-factorization. In the case of the squares being of even size, we use Hall’s matching theorem along with a $1$-factorization if $[\frac{n^2}{v}]$ is even, otherwise, Hall’s matching theorem together with Fulkerson’s~$[4]$ theorem, on the existence of a feasible flow in a network with bounds on flow leaving the sources and entering the sinks, lead to the required solution.