### On quadratic residue codes and hyperelliptic curves

*David Joyner*

#### 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.''Full Text: PDF PostScript