Weights in codes and genus 2 curves

McGuire, Gary; Voloch, José

doi:10.1090/S0002-9939-05-08027-5

Weights in codes and genus 2 curves
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by Gary McGuire and José Felipe Voloch

Proc. Amer. Math. Soc. 133 (2005), 2429-2437

DOI: https://doi.org/10.1090/S0002-9939-05-08027-5

Published electronically: March 15, 2005

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Abstract:

We discuss a class of binary cyclic codes and their dual codes. The minimum distance is determined using algebraic geometry and an application of Weil’s theorem. We relate each weight appearing in the dual codes to the number of rational points on a genus 2 curve of 2-rank 1 over a finite field of characteristic 2. The possible values for the number of points on a curve of genus 2 and 2-rank 1 are determined, thus determining the weights in the dual codes.

References

Yves Aubry and Marc Perret, A Weil theorem for singular curves, Arithmetic, geometry and coding theory (Luminy, 1993) de Gruyter, Berlin, 1996, pp. 1–7. MR 1394921
G. Cardona, E. Nart and J. Pujolas, Curves of genus two over fields of even characteristic. preprint. arXiv:math.NT/0210105.
Max Deuring, Die Typen der Multiplikatorenringe elliptischer Funktionenkörper, Abh. Math. Sem. Hansischen Univ. 14 (1941), 197–272 (German). MR 5125, DOI 10.1007/BF02940746
Gerhard Frey and Ernst Kani, Curves of genus $2$ covering elliptic curves and an arithmetical application, Arithmetic algebraic geometry (Texel, 1989) Progr. Math., vol. 89, Birkhäuser Boston, Boston, MA, 1991, pp. 153–176. MR 1085258, DOI 10.1007/978-1-4612-0457-2_{7}
Gerard van der Geer and Marcel van der Vlugt, Supersingular curves of genus $2$ over finite fields of characteristic $2$, Math. Nachr. 159 (1992), 73–81. MR 1237102, DOI 10.1002/mana.19921590106
Tadao Kasami, Weight distributions of Bose-Chaudhuri-Hocquenghem codes, Combinatorial Mathematics and its Applications (Proc. Conf., Univ. North Carolina, Chapel Hill, N.C., 1967) Univ. North Carolina Press, Chapel Hill, N.C., 1969, pp. 335–357. MR 0252100
Gilles Lachaud and Jacques Wolfmann, The weights of the orthogonals of the extended quadratic binary Goppa codes, IEEE Trans. Inform. Theory 36 (1990), no. 3, 686–692. MR 1053865, DOI 10.1109/18.54892
Daniel Maisner and Enric Nart, Abelian surfaces over finite fields as Jacobians, Experiment. Math. 11 (2002), no. 3, 321–337. With an appendix by Everett W. Howe. MR 1959745
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
René Schoof, Families of curves and weight distributions of codes, Bull. Amer. Math. Soc. (N.S.) 32 (1995), no. 2, 171–183. MR 1302786, DOI 10.1090/S0273-0979-1995-00586-0
J.-P. Serre, A course in arithmetic, Graduate Texts in Mathematics, No. 7, Springer-Verlag, New York-Heidelberg, 1973. Translated from the French. MR 0344216