Extremal numbers for cycles in a hypercube

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 $k≥3,$
$ex(Q_n,C_{4k+2})=O(n^{\frac{5}{6}+\frac{1}{3(2k−2)}}2^n)$.