MINIMAL AND CANONICAL RATIONAL GENERATOR MATRICES FOR CONVOLUTIONAL-CODES

A full-rank k x n matrix G(D) over the rational functions F(D) generat es a rate R = k/n convolutional code C. G(D) is minimal if it can be r ealized 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 s equences u(D), the span of u(D)G(D) covers the span of u(D). Alternati vely, G(D) is minimal if and only if G(D) is globally zero-free, or gl obally invertible. We show that G(D) is canonical if and only if G(D) is minimal and also globally orthogonal, in the valuation-theoretic se nse of Monna.