On quadratic residue codes and hyperelliptic curves

For an odd prime p and each non-empty subset S subset of GF (p), consider the hyperelliptic curve X-S defined by y(2) = f(S) (x), where f(S) (x) = Pi (a is an element of 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 subset of GF (p) for which the bound vertical bar X-S (GF (p))vertical bar > 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.".