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.