Every Convolutional Code is a Goppa Code
José Ignacio Iglesias Curto, Ángel Luis Muñoz Castañeda, José M. Muñoz Porras, Gloria Serrano Sotelo · IEEE Transactions on Information Theory · 2013
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.