Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)

 
 

 

Families of curves and weight distributions of codes


Author: René Schoof
Journal: Bull. Amer. Math. Soc. 32 (1995), 171-183
MSC: Primary 94B27; Secondary 11T71, 14H10, 94B15
DOI: https://doi.org/10.1090/S0273-0979-1995-00586-0
MathSciNet review: 1302786
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this expository paper we show how one can, in a uniform way, calculate the weight distributions of some well-known binary cyclic codes. The codes are related to certain families of curves, and the weight distributions are related to the distribution of the number of rational points on the curves.


References [Enhancements On Off] (What's this?)

  • [1] Atkin, A.O.L. and Lehner, J., Hecke operators on $ {\Gamma_{0}}(m)$, Math. Ann. 185 (1970), 134-160. MR 0268123 (42:3022)
  • [2] Deuring, M., Die Typen der Multiplikatorenringe elliptischer Funktionenkörper, Abh. Math. Sem. Hansischen Univ. 14 (1941), 197-272. MR 0005125 (3:104f)
  • [3] Cohen, H., Trace des opérateurs de Hecke sur $ {\Gamma_{0}}(N)$, Sém. Théorie Nombres 4 (1976-1977), Bordeaux. MR 0562292 (58:27771)
  • [4] Goppa, V.G., Codes on algebraic curves, Soviet Math. Dokl. 24 (1981), 170-172. MR 628795 (82k:94017)
  • [5] Kasami, T., Weight distributions of Bose-Chaudhuri-Hocquenghem codes (Bose, R.S. and Dowling, T.A., eds.), Combinatorial Math, and its Applications, Univ. of North Carolina Press, Chapel Hill, NC, 1969. MR 0252100 (40:5325)
  • [6] -, The weight enumerators for several classes of subcodes of the 2nd order binary Reed-Muller codes, Inform, and Control (Shenyang) 18, 369-394.
  • [7] Lachaud, G. and Wolfmann, J., The weights of the orthogonals of the extended quadratic binary Goppa codes, IEEE Trans. Inform. Theory 36 (1990), 686-692. MR 1053865 (92b:94040)
  • [8] MacWilliams, J. and Sloane, N.J.A., The theory of error-correcting codes, North Holland, Amsterdam, 1983.
  • [9] Melas, C.M., A cyclic code for double error correction, IBM J. Res. Develop. 4 (1960), 364-366. MR 0110594 (22:1470)
  • [10] Oesterlé, J., Sur la trace des opérateurs de Hecke, Thèse de $ {3^{\circ}}$ cycle, Orsay, 1977.
  • [11] Schoof, R., Non-singular plane cubic curves over finite fields, J. Combin. Theory Ser. A 46 (1987), 183-211. MR 914657 (88k:14013)
  • [12] Schoof, R. and Van der Vlugt, M., Hecke operators and the weight distributions of certain codes, J. Combin. Theory Ser. A 57 (1991), 163-186. MR 1111555 (92g:94017)
  • [13] Silverman, J., The arithmetic of elliptic curves, Graduate Texts in Math., vol. 106, Springer-Verlag, New York, 1986. MR 817210 (87g:11070)
  • [14] Van der Geer, G. and Van der Vlugt, M., Reed-Muller codes and supersingular curves. I, Compositio Math. 84 (1992), 333-367. MR 1189892 (93k:14038)
  • [15] Waterhouse, W., Abelian varieties over finite fields, Ann. Sci. École Norm. Sup. 2 (1969), 521-560. MR 0265369 (42:279)

Similar Articles

Retrieve articles in Bulletin of the American Mathematical Society with MSC: 94B27, 11T71, 14H10, 94B15

Retrieve articles in all journals with MSC: 94B27, 11T71, 14H10, 94B15


Additional Information

DOI: https://doi.org/10.1090/S0273-0979-1995-00586-0
Keywords: Weight distribution, coding theory, elliptic curve, Hecke operator
Article copyright: © Copyright 1995 American Mathematical Society

American Mathematical Society