Counting minimal generator matrices

Abstract

Given a particular convolutional code C, we wish to find all minimal generator matrices G(D) which represent that code. A standard form S(D) for a minimal matrix is defined, and then all standard forms for the code C are counted (this is equivalent to counting special pre-multiplication matrices P(D)). It is shown that all the minimal generator matrices G(D) are contained within the 'ordered row permutations' of these standard forms, and that all these permutations are distinct. Finally, the result is used to place a simple upper bound on the possible number of convolutional codes.

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