(A corrected preprint and an official version with a correction are available online.)

**Important note:** There is a mistake in Section 7 of the original version of this paper. A correction
can be found here. The correction has been incorporated into the
arXiv version of the paper.

We give new arguments that improve the known upper bounds on
the maximal number *N _{q}(g)* of rational points
of a curve of genus

To prove these results we make use of a number of Magma programs,
which we provide below.
Be warned that they have *not* been optimized. Also, the
programs may not work on versions of Magma earlier than 2.8.

- CheckQGN.magma.
Given a field size
*q*, a genus*g*, and a number*N*, this program will (eventually) list all of the possible real Weil polynomials of a genus-*g*curve over**F**_{q}having*N*points.**UPDATED ON 25 FEBRUARY 2007 to correct an error.**(See the changelog in the file for an explanation.) Fortunately, the error did not result in any mistakes in the published paper. - DeficientPolynomialList.magma. This file contains data that is used in the program CheckQGN.magma.
- Smyth.magma. This program allows us to recreate the tables that occur in a paper of Smyth.

- 3-6-15.magma. This is the
program we used to show that there are no genus-6 curves
over
**F**_{3}having exactly 15 points. - 27-4-66.magma. This is the
program we used to show that there are no genus-4 curves
over
**F**_{27}having exactly 66 points. - 27-4-65.magma. This is the
program we used to show that there are no genus-4 curves
over
**F**_{27}having exactly 65 points. - 32-4-75.magma. This is the
program we used to show that there are no genus-4 curves
over
**F**_{32}having exactly 75 points.

Our paper refers to the van der Geer-van der Vlugt tables of curves with many points. The version of their paper that we use is dated 18 January 2002, and we include a gzipped version of it here for reference. The current version can be found on Gerard van der Geer's web page.