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I am attempting to generate orbit representatives for the action of $\operatorname{GL}(n, F_2)$ on homogeneous polynomials of fixed degree $d$ in $n$ variables using random methods.

However, some polynomials have much smaller orbits than others, so many small orbits are proving elusive. Are there any efficient algorithmic methods for

  1. Generating homogeneous polynomials of fixed degree with prescribed symmetries, or more generally
  2. Generating random homogeneous polynomials of fixed degree that will weight heavily polynomials with small orbits/big stabilizers?

Generating examples in lexographic and various other orders is one way to catch small orbit elements, but I need a more general and better method.

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