\begin{block}{From Theory}
Find all the $2^n\times 2^n$ matrices $M=\dfrac{1}{n}.\hat{M}$ such that: 
\begin{enumerate}
\item $\hat{M}_{ij}=0$ if $j$ is not a neighbor of $i$
%, \textit{i.e.},  there is no edge from $i$ to $j$ in the $n$-cube.
\item $0 \le \hat{M}_{ii} \le n$: the number of loops around $i$ is lesser than $n$  
\item Otherwise $\hat{M}_{ij}=1$ if the edge from $i$ to $j$ is kept and 0 otherwise
\item For any index of line $i$, $1 \le i\le 2^n$, $n = \sum_{1 \le j\le 2^n} \hat{M}_{ij}$: 
  the matrix is right stochastic
\item For any index of column $j$, 
  $1 \le j\le 2^n$, $n = \sum_{1 \le i\le 2^n} \hat{M}_{ij}$: 
  the matrix is left stochastic 
\item All the values of $\sum_{1\le k\le 2^n}\hat{M}^k$ are strictly positive: the induced graph is strongly connected
\end{enumerate}
\end{block}



