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.