Definition (Binary Symmetric Channel)

This is a Discrete Memoryless Channel with $\mathcal{X}=\mathcal{Y}=\{0,1\}$ and $$P_{Y|X}(b|a)=\begin{cases}\epsilon&\mbox{if }a\not=b\\1-\epsilon&\mbox{if }a=b\end{cases} \ \ a,b\in\{0,1\}$$where $0\le\epsilon\le1$ is called the channel’s crossover probability or bit error rate. The transition matrix $Q$ is defined as $$Q=[P_{XY}]=\begin{bmatrix}P_{Y|X}(0|0)&P_{Y|X}(1|0)\\P_{Y|X}(0|1)&P_{Y|X}(1|1)\end{bmatrix}=\begin{bmatrix}1-\epsilon&\epsilon\\\epsilon&1-\epsilon\end{bmatrix}$$