The geometry of an interchange: Minimal matrices and circulants

An n × n zero-one matrix with constant column sums k is minimal if its determinant is ±k. A matrix having all line sums equal to k is minimal if its determinant ±k gcd(n, k). A general method is given for constructing minimal matrices using circulants. As a by-product, the interchange distance between a special circulant and the set of minimal matrices in its class is determined. Several open problems are stated.