On quadratic residue codes and hyperelliptic curves

For an odd prime p and each non-empty subset S ⊂ GF (p), consider the hyperelliptic curve XS defined by y = fS(x), where fS(x) = Q 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.”