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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
DOI: https://doi.org/10.1090/tran/7089
Published electronically: November 28, 2017
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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^{\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.


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Additional Information

Nathan Kaplan
Affiliation: Department of Mathematics, University of California, Irvine, California 92697-3875
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
Email: ian.petrow@math.ethz.ch

DOI: https://doi.org/10.1090/tran/7089
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

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