(A preprint is available online.)

For prime powers q<100, we compute new upper and lower bounds on N_q(4), the maximal number of points on a genus-4 curve over a finite field with q elements. We determine the exact value of N_q(4) for 17 prime powers q for which the value was previously unknown.

We use Magma to perform a number of calculations outlined in the paper. Our Magma programs can be found in two files:

• Genus4.magma
• NonMaximalOrders.magma

We also require a list of the isomorphism classes of unimodular quaternary Hermitian forms over the quadratic rings with discriminant -19 and -20; such lists are available from the Hermitian lattices web site maintained by Rainer Schulze-Pillot-Zieman. Copies of the relevant files, copied from the math.uni-sb.de server in March 2012, are available locally here:

• discriminant -3
• discriminant -4
• discriminant -7
• discriminant -19
• discriminant -20, type 1, even genus
• discriminant -20, type 1, odd genus
• discriminant -20, type 2