Do random linear codes satisfy the smooth min-entropy condition with high probability?

Ascertain whether a random linear outer code C ⊆ F_q^n (q a power of 2) of rate ε satisfies, with high probability, the per-codeword smoothed min-entropy property H_∞^{\bar{c}_η ε}(D_c) ≥ (1 − \bar{c}_γ)·log q for all nonzero c ∈ C, which would imply via Theorem 6 that C ∘ C approaches the GV bound.

Background

The second sufficient condition (Theorem 6) hinges on a smoothed min-entropy bound for empirical symbol distributions of all outer codewords. Whether random linear codes typically satisfy this stringent per-codeword property is unclear. A positive resolution would provide a direct probabilistic route to concatenated constructions near GV via the min-entropy method, yielding an alternate proof of the main existence result.