Soundness of the subspace inclusion test at the half-dimension regime

Prove that for the subspace-versus-subspace inclusion test used to encode linear functions, when the inner dimension is set to \ell_2 = \ell_1/2 (so the alphabet size is R = q^{\ell_1}), the test achieves soundness error \Theta(1/\sqrt{R}). Concretely, show that any two tables assigning linear functions to \ell_1- and \ell_2-dimensional subspaces that pass the inclusion test with probability \Omega(1/\sqrt{R}) must correspond to a legitimate global linear function agreeing on a significant fraction of the relevant subspaces.

Background

The paper studies a subspace encoding and an inclusion test: choose a random \ell_1-dimensional subspace L_1 and a random \ell_2-dimensional subspace L_2 \subseteq L_1, then check that the assigned functions agree on L_2. The authors analyze the achievable soundness in terms of the alphabet size R = q^{\ell_1}, aiming for the optimal \Theta(1/R) tradeoff.

In the natural half-dimension setting \ell_2 = \ell_1/2, a random strategy passes with probability \Theta(1/\sqrt{R}), suggesting \Theta(1/\sqrt{R}) soundness might be achievable. However, the current best analysis only proves \Theta(1/R^{1/4}), leaving a quadratic gap and motivating a precise soundness characterization.