On the Existence of Asymptotically Good Linear Codes in Minor-Closed Classes

Let C = (C₁, C₂, ...) be a sequence of codes such that each C_i is a linear [n_i, k_i, d_i]-code over some fixed finite field F, where n_i is the length of the code words, k_i is the dimension, and d_i is the minimum distance. We say that C is asymptotically good if, for some ε > 0 and for all i ∈ ℤ_>0, we have n_i≥ i and min(k_i/n_i, d_i/n_i) ≥ ε. Sequences of asymptotically good codes exist. We prove that if C is a class of GF(pⁿ)-linear codes (where p is prime and n ≥ 1), closed under puncturing and shortening, and if C contains an asymptotically good sequence, then C must contain all GF(p)-linear codes. Our proof relies on a powerful new result from matroid structure theory.