On quadratic residue codes and hyperelliptic curves

Abstract

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) = ∏ 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 |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.''