Feasibility of the smooth min-entropy condition for linear outer codes

Determine whether there exist linear outer codes C ⊆ F_q^n (q a power of 2) with rate ε such that for every nonzero codeword c ∈ C, the empirical symbol distribution D_c satisfies the smoothed min-entropy condition H_∞^{\bar{c}_η ε}(D_c) ≥ (1 − \bar{c}_γ)·log q for absolute constants \bar{c}_γ, \bar{c}_η > 0, as required in Theorem 6 to ensure that C ∘ C approaches the GV bound with high probability over a random inner code.

Background

The paper proposes a second sufficient condition (Theorem 6) for an outer code C that ensures the concatenation C ∘ C with a random linear inner code lies near the GV bound. This condition requires every nonzero outer codeword to have a symbol distribution with large smoothed min-entropy. The authors highlight that, unlike their first sufficient condition, they currently lack a feasibility proof for this smoothness requirement.