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.