Convolutional Goppa codes (CGC) were defined in Appl. Algebra Eng. Comm. Comput., vol. 15, pp. 51-61, 2004 and IEEE Trans. Inf. Theory, vol. 52, 340-344, 2006. In this paper, we prove that every convolutional code is a CGC defined over a smooth curve over$ \BBF _{q}(z)$and we give an explicit construction of convolutional codes as CGC over the projective line$ \BBP _{ \BBF _{q}(z)}^{1}$. We characterize which convolutional codes are defined by a complete linear system over curves of genus 0, 1, and over hyperelliptic curves. We apply these results to provide detailed constructions of some linear block codes as Goppa codes.

We show that the free distance, as a function on a space parameterizing a family of convolutional codes, is a lower semicontinuous function and that, therefore, the property of being maximum distance separable (MDS) is an open condition. For a class of convolutional codes, an algorithm is offered to compute the free distance. The behavior of the free distance by enlargements of the alphabet and by increasing the length is also studied. As an application, the algebraic equations characterizing the subfamily of MDS codes are explicitly computed for families of 1-D convolutional Goppa codes.

In this correspondence, we define convolutional Goppa codes over algebraic curves and construct their corresponding dual codes. Examples over the projective line and over elliptic curves are described, obtaining in particular some maximum-distance separable (MDS) convolutional codes.