Every Convolutional Code is a Goppa Code

Author

Iglesias Curto, Jose Ignacio ; Munoz Castaneda, Angel Luis ; Munoz Porras, Jose Maria ; Serrano Sotelo, Gloria

Author_Institution

Dept. of Math. & IUFFyM, Univ. de Salamanca, Salamanca, Spain

Volume

59

Issue

10

fYear

2013

fDate

Oct. 2013

Firstpage

6628

Lastpage

6641

Abstract

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.

Keywords

Goppa codes; block codes; convolutional codes; linear codes; CGC; convolutional Goppa code; explicit construction; hyperelliptic curve; linear block code; linear system; projective line; smooth curve; Block codes; Convolutional codes; Generators; Geometry; Linear systems; Terminology; Vectors; Algebraic curve; Goppa code; complete linear system; convolutional code; rational point;

fLanguage

English

Journal_Title

Information Theory, IEEE Transactions on

Publisher

ieee

ISSN

0018-9448

Type

jour

DOI

10.1109/TIT.2013.2271033

Filename

6547171