On the existence of asymptotically good linear codes in minor-closed classes

TL;DR

This paper proves that in minor-closed classes of linear codes over finite fields, the presence of an asymptotically good sequence implies the class contains all codes over the smallest field, using matroid theory.

Contribution

It establishes a structural characterization of minor-closed classes of linear codes containing asymptotically good sequences, linking code properties to matroid theory.

Findings

Asymptotically good codes imply the inclusion of all GF(p)-linear codes in certain classes.

Matroid structure theory is used to prove the main result.

The result applies to classes closed under puncturing and shortening.

Abstract

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.