Algebraic Geometry Constructions of Convolutional Codes

Abstract

Algebraic-geometric techniques to construct linear codes can be applied to construct convolutional codes, using algebraic curves over function fields. In this way we construct convolutional Goppa codes and provide a systematic way for constructing convolutional codes with prescribed properties. We study convolutional Goppa codes defined by the projective line and elliptic curves in detail.

Convolutional Goppa codes defined on fibrations

“…Although this may allow us to generalize the notion of convolutional code as subspaces of F q (z) n , one has to bear in mind that different associated encoders G generate submodules of F q [z] n which may be different. However, basic encoders always generate the same submodule ( [2]). Therefore, in this sense we may consider that the notions of convolutional codes as submodules of F q [z] n or as vector subspaces of F q (z) n are equivalent.…”

“…Indeed, for a given convolutional code C , the unimodular group GL(k, F q [z]) acts transitively on the set of basic encoders for C ( [2]). Then, one can consider an invariant associated with the code, the degree of the code, δ , defined as (e.g.…”

“…This paper offers a generalization of our algebro-geometric construction of Convolutional Goppa Codes (CGC) ( [1,2,9,10,14]). Recall, that Algebraic Geometry has been successfully applied in Coding Theory during the last decades, first for block codes (e.g.…”