Abstract
One of the most interesting class of curves, from the perspective of arithmetical algebraic geometry, are the so-called modular curves. Some of the most remarkable applications of algebraic geometry to coding theory arise from these modular curves. It turns out these algebraic-geometric codes (“AG codes”) constructed from modular curves can have parameters which beat the Gilbert–Varshamov lower bound if the ground field is sufficiently large.
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The expository paper [JS] discussed this in more detail from the computational perspective.
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The space \(\mathbb{H}=\{ z \in\mathbb{C}\;|\; \mathrm{Im}\,(z) > 0\}\) is also called the Poincaré upper half plane.
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Type optional_packages() for the name of the latest version of this database. This loads both ClassicalModularPolynomialDatabase and AtkinModularPolynomialDatabase.
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In fact, if we write \(f(z)=\,\sum _{n=1}^{\infty}a_{n}q^{n}\), then
$$\zeta_C(s)=\bigl(1-p^{-s}\bigr)^{-1}\prod_{p\not= 11}\bigl(1-a_pp^{-s}+p^{1-2s}\bigr)^{-1}$$is the global Hasse–Weil zeta function of the elliptic curve C of conductor 11 with Weierstrass model y 2+y=x 3−x 2 [Gel] (p. 252).
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In other words, C has length 56, dimension 22 over \({\mathbb{F}}\), and minimum distance 32.
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Recall Singleton’s bound: n≥d+k−1.
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Joyner, D., Kim, JL. (2011). Codes from Modular Curves. In: Selected Unsolved Problems in Coding Theory. Applied and Numerical Harmonic Analysis. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-8256-9_6
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