Minimal and canonical rational generator matrices for convolutional codes
A full-rank IC x n matrix G(D) over the rational functions F(D) generates a rate R = k/n convolutional code C. G(D) is minimal if it can be realized with as few memory elements as any encoder for C, and G(D) is canonical if it has a minimal realization in controller canonical form. We show that G(D) is minimal if and only if for all rational input sequences p1 (D), the span of U (D) G (D) covers t