Convolutional codes with additional algebraic structure

Convolutional codes have appeared in the literature endowed with sufficient additional algebraic structure to be considered as (left) ideals of a (code-ambient) automorphism-twisted polynomial ring with coefficients in a (word-ambient) semisimple finite group ring. In this paper we extend the present scope of the theory by considering a code-ambient twisted polynomial ring having, in addition to an automorphism σ, the action of a σ-derivation δ. In addition, we develop the basic theory without any specific restrictions for the semisimple finite word-ambient ring. This second element therefore extends even the original notions of both cyclic and group convolutional codes considered thus far in the literature. Among other results, in this paper we develop a matrix-based approach to the study of our extended notion of group convolutional codes (and therefore of cyclic convolutional codes as well), inspired by the use of circulant matrices by Gluesing-Luerssen and Schmale, and then use it to extend to this level the results on the existence of dual codes that were originally established by those authors for cyclic codes (in the narrower sense without a σ-derivation). Various examples illustrate the potential value of extending the search for good convolutional codes in this direction.