Everett W. Howe: New bounds on the maximum number of points on genus-4 curves over small finite fields, pp. 69–86 in: Arithmetic, Geometry, Cryptography and Coding Theory (Y. Aubry, C. Ritzenthaler, and A. Zykin, eds.), Contemporary Mathematics 574, American Mathematical Society, Providence, RI, 2012,

(A preprint is available online.)

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

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: