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."
Discrete Mathematics & Theoretical Computer Science