The paper proves that the order of the minimal state-space realisation of a set of irreducible implicit input--output (i/o) difference equations is equal to the degree of the Dieudonne determinant of a certain polynomial matrix. This matrix is defined by the generic linearisation of the i/o equations, and the map that incorporates the knowledge of the i/o equations in the polynomial coefficients. A simple algorithm to compute the degree of the Dieudonne determinant is given. Several examples demonstrate the theoretical result.