Codes and the Cartier operator

Couvreur, Alain

doi:10.1090/S0002-9939-2014-12011-9

Codes and the Cartier operator
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by Alain Couvreur

Proc. Amer. Math. Soc. 142 (2014), 1983-1996

DOI: https://doi.org/10.1090/S0002-9939-2014-12011-9

Published electronically: March 14, 2014

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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.

References

D. J. Bernstein, T. Lange, and C. Peters. Wild McEliece. Cryptology ePrint Archive, Report 2010/410, 2010. http://eprint.iacr.org/.
Wieb Bosma, John Cannon, and Catherine Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3-4, 235–265. Computational algebra and number theory (London, 1993). MR 1484478, DOI 10.1006/jsco.1996.0125
Pierre Cartier, Une nouvelle opération sur les formes différentielles, C. R. Acad. Sci. Paris 244 (1957), 426–428 (French). MR 84497
Pierre Cartier, Questions de rationalité des diviseurs en géométrie algébrique, Bull. Soc. Math. France 86 (1958), 177–251 (French). MR 106223, DOI 10.24033/bsmf.1503
V. D. Goppa, Codes on algebraic curves, Dokl. Akad. Nauk SSSR 259 (1981), no. 6, 1289–1290 (Russian). MR 628795
Tom Høholdt and Ruud Pellikaan, On the decoding of algebraic-geometric codes, IEEE Trans. Inform. Theory 41 (1995), no. 6, 1589–1614. Special issue on algebraic geometry codes. MR 1391018, DOI 10.1109/18.476214
Yasutaka Ihara, Some remarks on the number of rational points of algebraic curves over finite fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), no. 3, 721–724 (1982). MR 656048
G. L. Katsman and M. A. Tsfasman, A remark on algebraic geometric codes, Representation theory, group rings, and coding theory, Contemp. Math., vol. 93, Amer. Math. Soc., Providence, RI, 1989, pp. 197–199. MR 1003354, DOI 10.1090/conm/093/1003354
F. J. MacWilliams and N. J. A. Sloane, The theory of error-correcting codes. I, North-Holland Mathematical Library, Vol. 16, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. MR 0465509
C. S. Seshadri. L’opération de Cartier. Applications. In Variétés de Picard, volume 4 of Séminaire Claude Chevalley. Secrétariat Mathématiques, Paris, 1958-1959.
Alexei N. Skorobogatov, The parameters of subcodes of algebraic-geometric codes over prime subfields, Discrete Appl. Math. 33 (1991), no. 1-3, 205–214. Applied algebra, algebraic algorithms, and error-correcting codes (Toulouse, 1989). MR 1137746, DOI 10.1016/0166-218X(91)90116-E
Henning Stichtenoth, On the dimension of subfield subcodes, IEEE Trans. Inform. Theory 36 (1990), no. 1, 90–93. MR 1043283, DOI 10.1109/18.50376
Henning Stichtenoth, Algebraic function fields and codes, 2nd ed., Graduate Texts in Mathematics, vol. 254, Springer-Verlag, Berlin, 2009. MR 2464941
Yasuo Sugiyama, Masao Kasahara, Shigeichi Hirasawa, and Toshihiko Namekawa, Further results on Goppa codes and their applications to constructing efficient binary codes, IEEE Trans. Inform. Theory IT-22 (1976), no. 5, 518–526. MR 479662, DOI 10.1109/tit.1976.1055610
Michael Tsfasman, Serge Vlăduţ, and Dmitry Nogin, Algebraic geometric codes: basic notions, Mathematical Surveys and Monographs, vol. 139, American Mathematical Society, Providence, RI, 2007. MR 2339649, DOI 10.1090/surv/139
M. A. Tsfasman, S. G. Vlăduţ, and Th. Zink, Modular curves, Shimura curves, and Goppa codes, better than Varshamov-Gilbert bound, Math. Nachr. 109 (1982), 21–28. MR 705893, DOI 10.1002/mana.19821090103
Michael Wirtz, On the parameters of Goppa codes, IEEE Trans. Inform. Theory 34 (1988), no. 5, 1341–1343. Coding techniques and coding theory. MR 987679, DOI 10.1109/18.21264