On the Equivalence of Mann's Group Automorphism Method of Constructing an $O(n, n - 1)$ Set and Raktoe's Collineation Method of Constructing a Balanced Set of $L$-Restrictional Prime-Powered Lattice Designs

Abstract

This paper demonstrates the direct relationship which exists between $O(p^m, p^m - 1)$ sets and a balanced set of $\ell$-restrictional lattice designs for $p^m$ treatments. For instance, we will show that \begin{equation*}A = \begin{bmatrix}0 & 1 \\ 1 & 1\end{bmatrix}\end{equation*} which, as has been shown by Raktoe [5] completely specifies a balanced set of 2-restrictional lattice designs for $7^2 = 49$ treatments, will also completely characterize a set of 48 mutually orthogonal latin squares of order 49, i.e. an $O(49, 48)$ set. In other words, if our interest is to exhibit an $O$(49, 48) set, the above $2 \times 2$ matrix will do the job. Strangely enough, as will be shown, $A$ also completely characterizes an $O$(4, 3) set. Note that, since a balanced set of 1-restrictional lattice designs is simply a BIB design, this paper shows in particular a different proof for the known equivalence of the $O(p^m, p^m - 1)$ sets with a class of resolvable BIB designs. Consequently, the content of this paper will be useful for those who are concerned with tabulating the designs or writing an efficient program for generating designs on a computer.