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Codes from Jacobian surfaces. (English) Zbl 1369.94612

Bassa, Alp (ed.) et al., Arithmetic, geometry, cryptography and coding theory. 15th international conference on arithmetic, geometry, cryptography, and coding theory (AGCT), CIRM, Luminy, France, May 18–22, 2015. Proceedings. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-2810-5/pbk; 978-1-4704-3745-9/ebook). Contemporary Mathematics 686, 123-135 (2017).
Algebraic-geometry codes arising from algebraic curves defined over a finite field were proposed in the pioneer paper of V. D. Goppa [Dokl. Akad. Nauk SSSR 259, No. 6, 1289–1290 (1981; Zbl 0489.94014)]. These codes have pleasant properties and their parameters can be easily determined or bounded. In a similar way codes from higher dimensional varieties were proposed soon after.
The present paper gives a lower bound for the minimum distance of evaluation codes \(C(J_C,G)\), where \(J(C)\) is the Jacobian variety of a genus 2 curve \(C\) defined over a finite field \(\mathbb{F}_q\) and \(G\) is a very ample divisor numerically equivalent to \(rC\) for some positive integer \(r\). The proof is based on upper bounds for the number of points on irreducible curves on \(J(C)\).
Section 2 studies curves on abelian surfaces over finite fields and Theorem 2.4 provides a Weil type bound for the cardinal of curves lying on that abelian surface and in particular in \(J(C)\). Section 3 provides the wanted bound (Theorem 3.3): assuming \(J_C\) simple and \(r\leq (\mathrm{Card}(C(\mathbb{F}_q))/\lceil 2\sqrt{q}\rceil) -1 \) then \(d\geq \mathrm{Card}(J_C(\mathbb{F}_q))-r\mathrm{Card}(C(\mathbb{F}_q))\). Finally Section 4 gives several examples of codes \(C(J_C,G)\) showing the values of the lower bound and the true minimum distance (Table 1).
For the entire collection see [Zbl 1364.11004].

MSC:

94B27 Geometric methods (including applications of algebraic geometry) applied to coding theory
14G50 Applications to coding theory and cryptography of arithmetic geometry
14H10 Families, moduli of curves (algebraic)
14K12 Subvarieties of abelian varieties
11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
11G25 Varieties over finite and local fields

Citations:

Zbl 0489.94014
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References:

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