For g<=10 we construct the canonical curves via plane models.
For 10<g<14 the canonical curves are constructed via space models.
For g=14 the construction is based on Verra's proof of the unirationality of M_14 (see The unirationality of the moduli space of curves of genus ≤14 ).
The g=15 construction relies on matrix factorizations and is based on the Macaulay2 Package Matrix factorizations and families of curves of genus 15.
For g <=14, the methods used in this package are based on the Macaulay2 Package randomCurves and the methods for the g=15 case are based on the Macaulay2-package MatFac15.
This documentation describes version 0.3 of RandomCurvesOverVerySmallFiniteFields.
The source code from which this documentation is derived is in the file RandomCurvesOverVerySmallFiniteFields.m2.