Coefficient systems and supersingular representations of $GL_2(F)$

Let $F$ be a non-Archimedean local field with the residual characteristic $p$. We construct a "good" number of smooth irreducible $\bar{\mathbf{F}}_p$-representations of $GL_2(F)$, which are supersingular in the sense of Barthel and Livn\'e. If $F=\mathbf{Q}_p$ then results of Breuil imply that our construction gives all the supersingular representations up to the twist by an unramified quasi-character. We conjecture this is true for arbitrary $F$.