An invariant for matrices and sets of points in prime characteristic

Abstract

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.

Connections between Linear Complementary Dual Codes, Permanents and Geometry

“…For nonsquare matrices, there are quasi-invariants that are constructed, as in [11], from formulae involving subdeterminants of the matrix A. One of the authors in [12], for the cases n = k(q − 1), d = q − 1, q being any prime power, constructed invariants X q for prime characteristic fields. We shall see that there is a quasi-invariant that determines the binary LCD codes.…”