Prime power parallel flats fractions: Representation and properties
Abstract
A parallel flats fraction for the sn factorial experiment, where s = pm is a prime or prime power, is defined as the set of all solutions over GF(s) to At = ci, i = 1,2, . . . , f where A is r x n of rank r. In this paper we characterize the information matrix corresponding to a parallel flats fraction in terms of much smaller matrices P*j Pj over a commutative ring R for the case s = pm, m > 1. It is shown that the ring R is the direct k-sum of the cyclotomic field ℚ(ωp) where k = (pm - 1)/(p - 1). It is also shown that the matrices can be represented as k-tuples of matrices each over ℚ(ωp). The spectrum of the information matrix is obtained directly in terms of this representation in terms of k-tuples, as are additional properties such as the rank, determinant, and trace. Finally we derive a direct procedure for identifying the elements of the smaller matrices P*j Pj in terms of A and C. The result of this development is a very elegant method for determining the properties of parallel flats fractions At = C directly in terms of A and C. The theory is illustrated throughout with an example with s = 9 = 32, and a 93 parallel flats fraction.
