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

Authors: Gary McGuire and José Felipe Voloch
Journal: Proc. Amer. Math. Soc. 133 (2005), 2429-2437
MSC (2000): Primary 94B15, 11G20
DOI: https://doi.org/10.1090/S0002-9939-05-08027-5
Published electronically: March 15, 2005
MathSciNet review: 2138886
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

1. 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
2. G. Cardona, E. Nart and J. Pujolas, Curves of genus two over fields of even characteristic. preprint. arXiv:math.NT/0210105.
3. Max Deuring, Die Typen der Multiplikatorenringe elliptischer Funktionenkörper, Abh. Math. Sem. Hansischen Univ. 14 (1941), 197–272 (German). MR 0005125
4. 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
5. 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, https://doi.org/10.1002/mana.19921590106
6. 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
7. 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, https://doi.org/10.1109/18.54892
8. 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
9. F. J. MacWilliams and N. J. A. Sloane, The theory of error-correcting codes. I, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. North-Holland Mathematical Library, Vol. 16. MR 0465509
10. René Schoof, Families of curves and weight distributions of codes, Bull. Amer. Math. Soc. (N.S.) 32 (1995), no. 2, 171–183. MR 1302786, https://doi.org/10.1090/S0273-0979-1995-00586-0
11. J.-P. Serre, A course in arithmetic, Springer-Verlag, New York-Heidelberg, 1973. Translated from the French; Graduate Texts in Mathematics, No. 7. MR 0344216