Perfect codes and balanced generalized weighing matrices

It is proved that any set of representatives of the distinct one-dimensiona l subspaces in the dual code of the unique linear perfect single-error-corr ecting code of length (q(d) - 1)/(q - 1) over GF(q) is a balanced generaliz ed weighing matrix over the multiplicative group of GF(q). Moreover, this m atrix is characterized as the unique (up to equivalence) wieghing matrix fo r the given parameters with minimum q-rank. The classical, more involved co nstruction for this type of BGW-matrices is discussed for comparison, and a few monomially inequivalent examples are included. (C) 1999 Academic Press .