Perfect Codes and Balanced Generalized Weighing Matrices

Dieter Jungnickel
University of Augsburg, Germany
Joint work with V.D.Tonchev

Balanced generalized weighing matrices include well-known classical combinatorial objects such as Hadamard matrices and conference matrices. They have an interesting interpretation in Finite Geometries, and in this context they generalize notions like projective planes admitting a full elation or homology group. We will give some general background discussing these connections.

Then we will present a new construction method for BGW-matrices: Any set of representatives of the distinct 1-dimensional subspaces in the dual code of the unique linear perfect single-error-correcting code of length (q^d-1)/(q-1) over GF(q) is a balanced generalized weighing matrix over the multiplicative group of GF(q). Moreover, this matrix is characterized as the unique (up to equivalence) weighing matrix for the given parameters with minimum q-rank (namely d). We will describe the relation to the classical, more involved construction for this type of BGW-matrices (using affine geometry and relative difference sets). We 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.

Thus the talk will describe a topic which has aspects in several interesting areas of Discrete Mathematics: Design Theory, Finite Geometry, Galois Fields, and Coding Theory.