Traces of Hecke operators and refined weight enumerators of Reed-Solomon codes
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- by Nathan Kaplan and Ian Petrow PDF
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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 $q^{\mathrm {th}}$ 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
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Additional Information
- Nathan Kaplan
- Affiliation: Department of Mathematics, University of California, Irvine, California 92697-3875
- MR Author ID: 799489
- Email: nckaplan@math.uci.edu
- 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
- MR Author ID: 1027339
- ORCID: 0000-0001-8787-1703
- Email: ian.petrow@math.ethz.ch
- 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.
- © Copyright 2017 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 370 (2018), 2537-2561
- MSC (2010): Primary 11T71; Secondary 11F25, 11G20, 94B27
- DOI: https://doi.org/10.1090/tran/7089
- MathSciNet review: 3748576