Perfect Codes and Balanced Generalized Weighing Matrices

It is proved that any set of representatives of the distinct one-dimensional subspaces in the dual code of the unique linear perfect single-error-correcting code of length (qd−1)/(q−1) overGF(q) is a balanced generalized weighing matrix over the multiplicative group ofGF(q). Moreover, this matrix is characterized as the unique (up to equivalence) wieghing matrix for the given parameters with minimumq-rank. The classical, more involved construction for this type of BGW-matrices is discussed for comparison, and a few monomially inequivalent examples are included.