Asymptotic fraction of profile vectors captured by Varshamov AECC intersection

Prove that for any integers q, ℓ, and d, with S = {0,1,...,q−1}^ℓ and N = |S|, if (H, b) is a Varshamov-type (N, d+1) asymmetric error-correcting code over the finite field F_p that contains the all-ones vector, then the leading constant c(H, S) in the asymptotic expansion of |C(H, b) ∩ (n; S)| satisfies c(H, S) ≥ c(q, ℓ)/p^d, where c(q, ℓ) is the leading constant in the asymptotic expansion of |(n; S)| for S = {0,1,...,q−1}^ℓ.

Background

The paper defines c(q, ℓ) as the leading coefficient in the asymptotic formula |(n; S)| = c(q, ℓ) n^{|S|−|V(S)|} + O(n^{|S|−|V(S)|−1}) for S = {0,1,...,q−1}^ℓ, using Ehrhart theory for counting integer points in polytopes corresponding to profile vectors.

For a Varshamov-type (N, d+1) AECC C(H, b) over F_p that contains the all-ones vector and with N = |S|, the authors define c(H, S) as the leading coefficient in the asymptotic lower bound |C(H, b) ∩ (n; S)| ≥ c(H, S) n^{|S|−|V(S)|} + O(n^{|S|−|V(S)|−1}). The conjecture posits that asymptotically the intersection captures at least a 1/p^d fraction of all profile vectors, matching the naive pigeonhole expectation.