[Cherubino] Quaderni Elettronici del Seminario di Geometria Combinatoria 16E (Febbraio 2005), 1--39
Università degli Studi di Roma "La Sapienza" - Dipartimento di Matematica

Balanced Generalized Weighing Matrices and Related Structures
(Febbraio 2005)

Abstract. 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, and in this context they generalize notions like projective planes admitting a full elation or homology group. After explaining these basic connections in detail, we will restrict attention to 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 is usually 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.

There are also many applications of BGW-matrices to constructions of designs and graphs; we will restrict ourselves to just one construction method for symmetric designs due to Yury Ionin, for which we will work out an example in detail.

Dieter Jungnickel
Lehrstuhl fuer Diskrete Mathematik,
Optimierung und Operations Research
Universitaet Augsburg
D-86153 Augsburg, Germany
e-mail: jungnickel@math.uni-augsburg.de

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