Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Traces of Hecke operators and refined weight enumerators of Reed-Solomon codes

Authors: Nathan Kaplan and Ian Petrow
Journal: Trans. Amer. Math. Soc. 370 (2018), 2537-2561
MSC (2010): Primary 11T71; Secondary 11F25, 11G20, 94B27
Published electronically: November 28, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study the quadratic residue weight enumerators of the dual projective Reed-Solomon codes of dimensions $ 5$ and $ q-4$ over the finite field $ \mathbb{F}_q$. Our main results are formulas for the coefficients of the quadratic residue weight enumerators for such codes. If $ q=p^v$ and we fix $ v$ and vary $ p$, then our formulas for the coefficients of the dimension $ q-4$ code involve only polynomials in $ p$ and the trace of the $ q^{\textup {th}}$ and $ (q/p^2)^{\text {th}}$ Hecke operators acting on spaces of cusp forms for the congruence groups $ \operatorname {SL}_2 (\mathbb{Z}), \Gamma _0(2)$, and $ \Gamma _0(4)$. The main tool we use is the Eichler-Selberg trace formula, which gives along the way a variation of a theorem of Birch on the distribution of rational point counts for elliptic curves with prescribed $ 2$-torsion over a fixed finite field.

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

  • [1] Scott Ahlgren, The points of a certain fivefold over finite fields and the twelfth power of the eta function, Finite Fields Appl. 8 (2002), no. 1, 18-33. MR 1872789,
  • [2] B. J. Birch, How the number of points of an elliptic curve over a fixed prime field varies, J. London Math. Soc. 43 (1968), 57-60. MR 0230682,
  • [3] Bradley W. Brock and Andrew Granville, More points than expected on curves over finite field extensions, Finite Fields Appl. 7 (2001), no. 1, 70-91. MR 1803936,
  • [4] David A. Cox, Primes of the form $ x^2 + ny^2$: Fermat, class field theory, and complex multiplication, 2nd ed., Pure and Applied Mathematics (Hoboken), John Wiley & Sons, Inc., Hoboken, NJ, 2013. MR 3236783
  • [5] Max Deuring, Die Typen der Multiplikatorenringe elliptischer Funktionenkörper, Abh. Math. Sem. Hansischen Univ. 14 (1941), 197-272 (German). MR 0005125
  • [6] Bas Edixhoven and Jean-Marc Couveignes (eds.), Computational aspects of modular forms and Galois representations: How one can compute in polynomial time the value of Ramanujan's tau at a prime, Annals of Mathematics Studies, vol. 176, Princeton University Press, Princeton, NJ, 2011. MR 2849700
  • [7] Martin Eichler, Quaternäre quadratische Formen und die Riemannsche Vermutung für die Kongruenzzetafunktion, Arch. Math. 5 (1954), 355-366 (German). MR 0063406,
  • [8] N. D. Elkies,
    Linear codes and algebraic geometry in higher dimensions.
    Preprint, 2006.
  • [9] Gerard van der Geer, René Schoof, and Marcel van der Vlugt, Weight formulas for ternary Melas codes, Math. Comp. 58 (1992), no. 198, 781-792. MR 1122080,
  • [10] Norman E. Hurt, Exponential sums and coding theory: a review, Acta Appl. Math. 46 (1997), no. 1, 49-91. MR 1432471,
  • [11] Yasutaka Ihara, Hecke polynomials as congruence $ \zeta$ functions in elliptic modular case, Ann. of Math. (2) 85 (1967), 267-295. MR 0207655,
  • [12] Nicholas M. Katz and Peter Sarnak, Random matrices, Frobenius eigenvalues, and monodromy, American Mathematical Society Colloquium Publications, vol. 45, American Mathematical Society, Providence, RI, 1999. MR 1659828
  • [13] Andrew Knightly and Charles Li, Traces of Hecke operators, Mathematical Surveys and Monographs, vol. 133, American Mathematical Society, Providence, RI, 2006. MR 2273356
  • [14] Gilles Lachaud, The parameters of projective Reed-Muller codes, Discrete Math. 81 (1990), no. 2, 217-221 (English, with French summary). MR 1054981,
  • [15] H. W. Lenstra Jr., Factoring integers with elliptic curves, Ann. of Math. (2) 126 (1987), no. 3, 649-673. MR 916721,
  • [16] Rudolf Lidl and Harald Niederreiter, Introduction to finite fields and their applications, 1st ed., Cambridge University Press, Cambridge, 1994. MR 1294139
  • [17] John B. Little, Algebraic geometry codes from higher dimensional varieties, Advances in algebraic geometry codes, Ser. Coding Theory Cryptol., vol. 5, World Sci. Publ., Hackensack, NJ, 2008, pp. 257-293. MR 2509126,
  • [18] 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
  • [19] Carlos Moreno, Algebraic curves over finite fields, Cambridge Tracts in Mathematics, vol. 97, Cambridge University Press, Cambridge, 1991. MR 1101140
  • [20] Gabriele Nebe, Eric M. Rains, and Neil J. A. Sloane, Self-dual codes and invariant theory, Algorithms and Computation in Mathematics, vol. 17, Springer-Verlag, Berlin, 2006. MR 2209183
  • [21] René Schoof, Nonsingular plane cubic curves over finite fields, J. Combin. Theory Ser. A 46 (1987), no. 2, 183-211. MR 914657,
  • [22] René Schoof, Families of curves and weight distributions of codes, Bull. Amer. Math. Soc. (N.S.) 32 (1995), no. 2, 171-183. MR 1302786,
  • [23] René Schoof and Marcel van der Vlugt, Hecke operators and the weight distributions of certain codes, J. Combin. Theory Ser. A 57 (1991), no. 2, 163-186. MR 1111555,
  • [24] Anders Bjært Sørensen, Projective Reed-Muller codes, IEEE Trans. Inform. Theory 37 (1991), no. 6, 1567-1576. MR 1134296,
  • [25] M. A. Tsfasman and S. G. Vlăduţ, Algebraic-geometric codes, Mathematics and its Applications (Soviet Series), vol. 58, Kluwer Academic Publishers Group, Dordrecht, 1991. MR 1186841
  • [26] William C. Waterhouse, Abelian varieties over finite fields, Ann. Sci. École Norm. Sup. (4) 2 (1969), 521-560. MR 0265369

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 11T71, 11F25, 11G20, 94B27

Retrieve articles in all journals with MSC (2010): 11T71, 11F25, 11G20, 94B27

Additional Information

Nathan Kaplan
Affiliation: Department of Mathematics, University of California, Irvine, California 92697-3875

Ian Petrow
Affiliation: École Polytechnique Fédérale de Lausanne, Section des Mathématiques, 1015 Lausanne, Switzerland
Address at time of publication: Departement Mathematik, ETH Zürich, HG G 66.4 Rämistrasse 101, 8092 Zürich, Switzerland

Received by editor(s): September 25, 2015
Received by editor(s) in revised form: July 15, 2016
Published electronically: November 28, 2017
Additional Notes: The second author was partially supported by Swiss National Science Foundation grant 200021-137488 and an AMS-Simons Travel Grant.
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society