Minimal and canonical rational generator matrices for convolutional codes

A full-rank K×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 u(D), the span of u(D)G(D) covers the span of u(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 (1970)