Extend the enumeration range for alternating matrix spaces

Relax the restriction that the dimension of an alternating matrix space be linearly related to the ambient dimension, so that the asymptotic enumeration result for congruence orbits of alternating matrix spaces applies beyond the range m=\Theta(n).

Background

The paper proves an asymptotic formula for the number of congruence orbits of m-dimensional subspaces of the space \Lambda(n,q) of n\times n alternating matrices over F_q when m=\lceil Cn+R\rceil, with fixed rational constants C>0 and R. This condition restricts the matrix-space dimension to be linear in n. The authors explicitly identify extending the result to a broader range of m as an open problem, in analogy with the wider parameter ranges known for enumerating unlabelled graphs.

References

Note that compared to Theorem~\ref{thm:wright}, the range $m=\Theta(n)$ in Corollary~\ref{thm:space} is restrictive, and it is an interesting open problem to relax it.

— On average orders of automorphism groups of bilinear maps over finite fields  (2503.07299 - Bläser et al., 10 Mar 2025) in Section 1, Application 2: random matrix spaces with trivial symmetry, immediately after Theorem 3 (Theorem~\ref{thm:space})