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 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.