Balanced generalized weighing matrices and their applications

Balanced generalized weighing matrices include well-known classical combinatorial objects such as Hadamard matrices and conference matrices; moreover, particular classes of BGW-matrices are equiva-lent to certain relative difference sets. BGW-matrices admit an interesting geometrical interpretation, and in this context they generalize notions like projective planes admitting a full elation or homology group. After surveying these basic connections, we will focus attention on proper BGW-matrices; thus we will not give any systematic treatment of generalized Hadamard matrices, which are the subject of a large area of research in their own right. In particular, we will discuss what might be called the classical parameter series. Here the nicest examples are closely related to per-fect codes and to some classical relative difference sets associated with affine geometries; moreover, the matrices in question can be charac-terized as the unique (up to equivalence) BGW-matrices for the given parameters with minimum q-rank. One can also obtain a wealth of monomially inequivalent examples and determine the q-ranks of all these matrices by exploiting a connection with linear shift register sequences. The final section of this survey will consider applications to con-structions of designs and graphs. We will divide this section into six parts. The first of these will deal with the work of Ionin. The second part will be devoted to Bush–type Hadamard matrices and twin designs. In the third part we will discuss the productive regular Hadamard matrices and symmetric designs. In the fourth part appli-cations to constructing strongly regular graphs are given. The fifth part concerns doubly regular digraphs, and the final part deals with some newly emerging applications