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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
Published electronically:
March 15, 2005
MathSciNet review:
2138886
Full-text PDF Free Access
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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.
- 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
(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.
Max
Deuring, Die Typen der Multiplikatorenringe elliptischer
Funktionenkörper, Abh. Math. Sem. Hansischen Univ.
14 (1941), 197–272 (German). MR 0005125
(3,104f)
- 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
(91k:14014)
- 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 (94g:11043), http://dx.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
(40 #5325)
- 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
(92b:94040), http://dx.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
(2003k:14057)
- 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
(57 #5408a)
- 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
(95j:94025), http://dx.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
(49 #8956)
- 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)
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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:
http://dx.doi.org/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
Published electronically:
March 15, 2005
Communicated by:
Wen-Ching Winnie Li
Article copyright:
© Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
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