Let us consider the Markov chain $(X_n)_{n \in \mathbb{N}}$ with state space $I = \{0,1\}^m$ and transition probabilities $$ p_{xy} = \begin{cases} m^{-1} &\mbox{if } \vert x - y \vert = 1 \\ 0 & \mbox{otherwise } \end{cases} $$
Here is $\vert x \vert = \sum_{k=1}^{m} \vert x_k \vert $.
I see the Markov chain as an $m$-dimensional unit hypercube and at each step from any vertex we have an equal probability of moving to any of the adjacent vertices.
The goal of this homework is to compute the invariant distribution of $X$. I know that we need the transition matrix and here I got stuck. I don't know how to assemble this matrix. Can anyone give some advice?