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 \(k \ge \left\lfloor\frac{3}{7}n\right\rfloor +1\).