For an odd prime p and each non-empty subset S ⊂ GF(p), consider the hyperelliptic curve X_S deﬁned by y^2 = f_s(x), where f_s(x) = \P_{a2S} (x-a). Using a connection between binary quadratic residue codes and hyperelliptic curves over GF(p), this paper investigates how coding theory bounds give rise to bounds such as the following example: for all sufﬁciently large primes p there exists a subset S ⊂ GF(p) for which the bound |X_S(GF(p))| > 1.39p holds. We also use the quasi-quadratic residue codes deﬁned below to construct an example of a formally self-dual optimal code whose zeta function does not satisfy the "Riemann hypothesis."

Source : oai:HAL:hal-00972302v1

Volume: Vol. 10 no. 1

Section: Combinatorics

Published on: January 1, 2008

Submitted on: March 26, 2015

Keywords: [INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]

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