|
Weights in codes and genus 2 curves
Author(s):
Gary
McGuire;
José
Felipe
Voloch
Journal:
Proc. Amer. Math. Soc.
133
(2005),
2429-2437.
MSC (2000):
Primary 94B15, 11G20
Posted:
March 15, 2005
Retrieve article in:
PDF DVI PostScript
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.
References:
-
- 1.
- Y. Aubry and M. Perret, A Weil theorem for singular curves, ``Arithmetic, geometry and coding theory (Luminy, 1993),'' 1-7, de Gruyter, Berlin, 1996. MR 1394921 (97g:11061)
- 2.
- G. Cardona, E. Nart and J. Pujolas, Curves of genus two over fields of even characteristic. preprint. arXiv:math.NT/0210105.
- 3.
- M. Deuring, Die Typen der Multiplicatorenringe elliptischer Functionenkörper. Hamburger Abhandlungen 14 (1941) 197-272. MR 0005125 (3:104f)
- 4.
- G. Frey and E. Kani, Curves of genus
 covering elliptic curves and an arithmetical application. in ``Arithmetic algebraic geometry (Texel, 1989),'' 153-176, Progr. Math., 89, Birkhäuser Boston, Boston, MA, 1991. MR 1085258 (91k:14014) - 5.
- G. van der Geer, M. van der Vlugt, Supersingular Curves of Genus 2 over Finite Fields of characteristic 2, Math. Nachr. 159 (1992) 73-81. MR 1237102 (94g:11043)
- 6.
- T. Kasami, Weight distributions of Bose-Chaudhuri-Hocquenghem codes, in ``Combinatorial Mathematics and its Applications (Proc. Conf., Univ. North Carolina, Chapel Hill, N.C., 1967),'' 335-357, Univ. North Carolina Press, Chapel Hill, N.C., 1969. MR 0252100 (40:5325)
- 7.
- G. Lachaud and J. Wolfmann, The weights of the orthogonals of the extended quadratic binary Goppa codes, IEEE Trans. Info. Th. 36 No. 3 (1990) 686-692. MR 1053865 (92b:94040)
- 8.
- D. Maisner and E. Nart, Abelian surfaces over finite fields as Jacobians, Experimental Math. 11 No. 3 (2002) 321-338. MR 1959745 (2003k:14057)
- 9.
- F. J. MacWilliams and N. J. A. Sloane, ``The Theory of Error-Correcting Codes,'' North Holland, Amsterdam, 1977. MR 0465509 (57:5408a)
- 10.
- R. Schoof, Families of curves and weight distributions of codes, Bull. AMS 32 No. 2 (1995) 171-183. MR 1302786 (95j:94025)
- 11.
- J.-P. Serre, ``A Course in Arithmetic,'' GTM 7, Springer-Verlag, New York, 1973. MR 0344216 (49:8956)
Similar Articles:
Retrieve articles in Proceedings of the American Mathematical Society
with MSC
(2000):
94B15, 11G20
Retrieve articles in all Journals with MSC
(2000):
94B15, 11G20
Additional Information:
Gary
McGuire
Affiliation:
Department of Mathematics, National University of Ireland, Maynooth, Co. Kildare, Ireland
José
Felipe
Voloch
Affiliation:
Department of Mathematics, University of Texas, Austin, Texas 78712
DOI:
10.1090/S0002-9939-05-08027-5
PII:
S 0002-9939(05)08027-5
Received by editor(s):
May 19, 2003
Received by editor(s) in revised form:
February 6, 2004
Posted:
March 15, 2005
Communicated by:
Wen-Ching Winnie Li
Copyright of article:
Copyright
2005,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
|
Comments: webmaster@ams.org