On quadratic residue codes and hyperelliptic curves

David Joyner
2008 Discrete Mathematics and Theoretical Computer Science DMTCS   unpublished
For an odd prime p and each non-empty subset S ⊂ GF (p), consider the hyperelliptic curve XS defined by . 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
more » ... le of a formally self-dual optimal code whose zeta function does not satisfy the "Riemann hypothesis." Keywords: binary linear codes, hyperelliptic curves over a finite field, quadratic residue codes, (11T71, 11T24, 14G50, 94B40, 94B27)