Researchers studying the theory of error-correcting codes have discovered, in recent years, that finite geometries and designs can provide the basis for excellent communications schemes. The basic idea is to take the linear span (over some finite field) of the rows of the incidence matrix of such a structure as the allowable messages. Mariner 9, for example, transmitted data to Earth by using a code derived from the structure of the hyperplanes in a five-dimensional vector space over F2 the field with two elements.

The purpose of this monograph is to allow coding theory to repay some of its debt to the combinatorial theory of designs. Specifically, I have tried to show herein how the objects introduced by coding theorists can offer great insight into the study of symmetric designs.

The vector spaces and modules (over appropriate rings) generated by the incidence matrices of symmetric designs provide a natural setting for invoking much algebraic machinery– – most notably, the theory of group representations– –which has hitherto not found much application in this combinatorial subject. In doing so, they provide a point of view which unifies a number of diverse results as well as makes possible many new theorems. My own investigation into this subject is surely not definitive, and if anyone is stimulated to further develop this point of view, I will have accomplished something.

Two goals have informed my choice of organization. First, since my object is to expose a particular approach to the study of symmetric designs, I have chosen to develop the subject from scratch.