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Codes and the Cartier operator


Author: Alain Couvreur
Journal: Proc. Amer. Math. Soc. 142 (2014), 1983-1996
MSC (2010): Primary 11G20, 14G50, 94B27
DOI: https://doi.org/10.1090/S0002-9939-2014-12011-9
Published electronically: March 14, 2014
MathSciNet review: 3182017
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Abstract: In this article, we present a new construction of codes from algebraic curves. Given a curve over a non-prime finite field, the obtained codes are defined over a subfield. We call them Cartier codes since their construction involves the Cartier operator. This new class of codes can be regarded as a natural geometric generalisation of classical Goppa codes. In particular, we prove that a well-known property satisfied by classical Goppa codes extends naturally to Cartier codes. We prove general lower bounds for the dimension and the minimum distance of these codes and compare our construction with a classical one: the subfield subcodes of Algebraic Geometry codes. We prove that every Cartier code is contained in a subfield subcode of an Algebraic Geometry code and that the two constructions have similar asymptotic performances.

We also show that some known results on subfield subcodes of Algebraic Geometry codes can be proved nicely by using properties of the Cartier operator and that some known bounds on the dimension of subfield subcodes of Algebraic Geometry codes can be improved thanks to Cartier codes and the Cartier operator.


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Additional Information

Alain Couvreur
Affiliation: INRIA Saclay Île-de-France – CNRS LIX, UMR 7161, École Polytechnique, 91128 Palaiseau Cedex, France
Email: alain.couvreur@lix.polytechnique.fr

DOI: https://doi.org/10.1090/S0002-9939-2014-12011-9
Keywords: Algebraic Geometry codes, differential forms, Cartier operator, subfield subcodes, classical Goppa codes
Received by editor(s): June 21, 2012
Received by editor(s) in revised form: July 23, 2012
Published electronically: March 14, 2014
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.