Definition (Binary erasure channel)

a DMC with alphabets $\mathcal{X}=\{0,1\}$, $\mathcal{Y}=\{0,E,1\}$. In the BSC, bits are received either perfectly or corrupted. Here we can also lose bits, (we know the location of the lost bits but not their value), these are denoted using $E$. We define the transition probability as $$P_{Y|X}(b|a)=\begin{cases}1-\alpha&\mbox{if }a=b,b\in\{0,1\}\\0&\mbox{if }a\not=b,a,b\in\{0,1\}\\\alpha&b=E,a\in\{0,1\}\end{cases}$$where $0\le\alpha\le1$ is the channel’s erasure probability. The transition matrix is defined as $$Q=[P_{XY}]=\begin{bmatrix}P_{Y|X}(0|0)&P_{Y|X}(E|0)&P_{Y|X}(1|0)\\P_{Y|X}(0|1)&P_{Y|X}(E|1)&P_{Y|X}(1|1)\end{bmatrix}=\begin{bmatrix}1-\alpha&\alpha&0\\0&\alpha&1-\alpha\end{bmatrix}$$