Minimal and canonical rational generator matrices for convolutional codes

In English 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 the span of ZL (D). Alternatively, G(D) is minimal if and only if G(D) is globally zero-free, or globally invertible. We show that G(D) is canonical if and only if G(D) is minimal and also globally orthogonal, in the valuation-theoretic sense of Monna.