  Let $\mathcal{C} = (C_1, C_2, \ldots)$ 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
$\mathbb{F}$, where $n_i$ is the length of the codewords, $k_i$ is the
dimension, and $d_i$ is the minimum distance. We say that $\mathcal{C}$ is
asymptotically good if, for some $\varepsilon > 0$ and for all $i$, $n_i \geq
i$, $k_i/n_i \geq \varepsilon$, and $d_i/n_i \geq \varepsilon$. Sequences of
asymptotically good codes exist. We prove that if $\mathcal{C}$ is a class of
GF$(p^n)$-linear codes (where $p$ is prime and $n \geq 1$), closed under
puncturing and shortening, and if $\mathcal{C}$ contains an asymptotically good
sequence, then $\mathcal{C}$ must contain all GF$(p)$-linear codes. Our proof
relies on a powerful new result from matroid structure theory.
