An invariant for matrices and sets of points in prime characteristic

There is polynomial function X _q in the entries of an m × m(q - 1) matrix over a field of prime characteristic p, where q = p ^h is a power of p, that has very similar properties to the determinant of a square matrix. It is invariant under multiplication on the left by a non-singular matrix, and under permutations of the columns. This gives a way to extend the invariant theory of sets of points in projective spaces of prime characteristic, to make visible hidden structure. There are connections with coding theory, permanents, and additive bases of vector spaces.