*Balanced Generalized Weighing Matrices and their
Applications*

Dieter Jungnickel, Universität Augsburg, Germany

Let G be a multiplicatively written group, and let 0 be a symbol not
contained in G. A *balanced generalized weighing matrix*
BGW(m,k,μ) over G is an m x m matrix W=(w_{ij}) with entries
from G'=G ∪ {0} satisfying the following two
conditions:

1) Each column of W contains exactly k nonzero entries.

2) For all a,b in {1,...,m} with a not equal b, the multiset
{w_{ai} w_{bi}^{-1} : 1 ≤ i ≤ m,
w_{ai}, w_{bi}≠ 0} contains exactly μ/|G|
copies of each element of G.

If there are no entries 0 (that is, for k=m) one speaks of a
*generalized Hadamard matrix* of order m over G and uses the notation
GH(n,λ), where n=|G| and λ=m/n. Finally, any BGW(n+1,n,n-1) is
called a *generalized conference matrix*.

As the terminology suggests, balanced generalized weighing matrices include well-known classical combinatorial objects such as Hadamard matrices and conference matrices; moreover, particular classes of BGW-matrices are equivalent to certain relative difference sets. BGW-matrices admit an interesting geometrical interpretation as class regular symmetric divisible designs, and in this context they generalize notions like projective planes admitting a full elation or homology group. After explaining these basic connections in detail, I will focus attention on proper BGW-matrices; thus I 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, I shall consider what is often called the *classical*
parameter series. Here the nicest examples are closely related to perfect
codes and to some classical relative difference sets associated with
affine geometries; moreover, the matrices in question can be characterized
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.

Finally, I shall also discuss some applications to the construction of (symmetric) designs and (various types of) graphs.