Generalized First order Descriptions and Canonical Forms for Convolutional Codes

A convolutional code C can be naturally viewed as a submodule of the free module F[z]. In this way we can view a convolutional code as dual to a linear, shift invariant and complete behavior. The known representations of linear behaviors carry over by duality to representations for convolutional codes. In the behavioral literature one can find two natural generalized first order forms and we refer to [3] for details. In this paper we study one of these generalized first order representations and we work out in a direct way the minimality conditions as well as the uniqueness of the description. Since the description is only unique up to a group action it is natural to study certain canonical forms of this first order description. In a last part of the paper we explain how these different canonical forms were recently useful in the construction of new convolutional codes with a designed free distance.