A Method of Constructing Partially Balanced Incomplete Block Designs

Abstract

Partially balanced incomplete block designs were introduced by Bose and Nair [1], who described a number of methods of constructing such designs. Among these methods there is one based on incidence properties of finite geometries. This uses the finite geometries associated with the Galois field $GF(p^n)$ with addition and multiplication $(\operatorname{mod} p$). By weakening the geometrical structure (or, equivalently, by weakening the rules of addition and multiplication), it is possible to obtain new designs. A basic feature of a finite projective geometry is that the coordinates are elements of a finite field. What we do here is to allow the coordinates to belong instead to a linear associative algebra $\mathscr{a},$ of finite order $n$ and with modulus, over a finite field $F.$ The procedure is summarized below and explained with more detail in regard to two designs. (For accounts of a similar geometrical theory, using an infinite field, see [7], [8], [9].)