On generalized Hadamard matrices of minimum rank

Generalized Hadamard matrices of order qn−1 (q—a prime power, n 2) over GF(q) are related to symmetric nets in affine 2-(qn,qn−1,(qn−1−1)/(q−1)) designs invariant under an elementary abelian group of order q acting semi-regularly on points and blocks. The rank of any such matrix over GF(q) is greater than or equal to n−1. It is proved that a matrix of minimum q-rank is unique up to a monomial equivalence, and the related symmetric net is a classical net in the n-dimensional affine geometry AG(n,q).