On the MDS conjecture in even characteristic

We prove that every nontrivial linear $[n,k]$ maximum distance separable (MDS) code over a finite field ${\text{F}}_q$ of characteristic two satisfies $n\le q+1$, except in dimensions $k=3$ and $k=q-1$, where $n\le q+2$. This resolves the linear MDS conjecture in even characteristic.