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 XS defined by
y2=fS(x),
where fS(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 |XS(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