Discrete Mathematics & Theoretical Computer Science, Vol 10, No 1 (2008)

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On quadratic residue codes and hyperelliptic curves

David Joyner


For an odd prime p and each non-empty subset S⊂GF(p), consider the hyperelliptic curve XS defined by y2=fS(x), where fS(x) = ∏a∈S(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 |XS(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.''

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