Geelen–Gerards–Whittle conjecture for minor-closed classes of binary codes

Let $\mathcal{C}$ be a proper subclass of the binary linear codes that is closed under puncturing and shortening. Let $\mathcal{G}$ denote the class of cycle codes, and let $\theta_{\mathcal{C}}$ and $\theta_{\mathcal{G}}$ be the corresponding threshold functions.

Geelen–Gerards–Whittle conjecture. Either $\mathcal{G} \subseteq \mathcal{C}$ and

$$\theta_{\mathcal{C}} = \theta_{\mathcal{G}},$$

or

$$\theta_{\mathcal{C}} = 0.$$

This conjecture predicts that every proper minor-closed subclass of binary linear codes has the threshold function of the class of cycle codes when it contains all cycle codes, and otherwise has identically zero threshold. It is motivated by the structural theory of minor-closed classes, according to which highly connected codes in such a class are close to cycle codes or their duals.