  Let $ex(Q_n, H)$ be the largest number of edges in a subgraph $G$ of a
hypercube $Q_n$ such that there is no subgraph of $G$ isomorphic to $H$. We
show that for any integer $k\geq 3$, $$ex(Q_n, C_{4k+2})= O(n^{\frac{5}{6} +
\frac{1}{3(2k-2)}} 2^n).$$
