Application of Cyclic Collineations to the Construction of Balanced $L$-Restrictional Prime Powered Lattice Designs

Abstract

The main problem encountered with any $s^m$ lattice design, where $s(= p^n)$ is a prime positive integer, is the construction of a balanced lattice so that after ordering the experimenter may select a best set of arrangements. When $s$ and $m$ are large the usual method of construction becomes quite laborious. The object of this paper is to develop a method of obtaining a balanced set of arrangements by means of cyclic collineations on the finite projective geometry $PG(k, s)$, where $k = m - 1$. Considerations are to be limited to collineations whose characteristic matrices $A (\rho) = A - \rho I$ have a single non-trivial invariant factor. The Smith canonical form of $A(\rho)$ is then diagonal $(1, 1, \cdots, 1, f_m(\rho))$ and thus we can limit consideration to the associated rational canonical form. Such matrices can be generated and the orders found by electronic computers for any $s^m$ lattice. It is shown that with proper choice of $GF(s)$ any balanced $l$-restrictional $s^m$ lattice is given by $\alpha = \sum^{m - 1}_{i = 0} s^i$ arrangements. Associating with the $\alpha$ arrangements the $\alpha$ powers of a cyclic collineation in rational canonical form of order $\alpha$ it is shown that the generators of the confounding scheme in each arrangement can immediately be taken from the columns of the respective powers of the matrix of the cyclic collineation. Balanced arrangements for lattices with $s^m < 1000$ are typified by presenting the associated cyclic collineations of order $\alpha$