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

Let C = (C
₁
, C
₂
, ...) 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(p
ⁿ
)-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.