Pairing strategies for the Maker-Breaker game on the hypercube with subcubes as winning sets

We consider the Maker-Breaker positional game on the vertices of the $n$-dimensional hypercube $\{0,1\}^n$ with $k$-dimensional subcubes as winning sets. We describe a pairing strategy which allows Breaker to win if $n$ is a power of 4 and $k \ge n/4 +1$. Our results also imply that for all $n \geq 3$ there is a Breaker's win pairing strategy if $k \ge \left\lfloor\frac{3}{7}n\right\rfloor +1$.