# Binary erasure channel

The channel model for the binary erasure channel showing a mapping from channel input X to channel output Y (with known erasure symbol ?). The probability of erasure is $\displaystyle{ p_e }$

In coding theory and information theory, a binary erasure channel (BEC) is a communications channel model. A transmitter sends a bit (a zero or a one), and the receiver either receives the bit correctly, or with some probability $\displaystyle{ P_e }$ receives a message that the bit was not received ("erased") .

## Definition

A binary erasure channel with erasure probability $\displaystyle{ P_e }$ is a channel with binary input, ternary output, and probability of erasure $\displaystyle{ P_e }$. That is, let $\displaystyle{ X }$ be the transmitted random variable with alphabet $\displaystyle{ \{0,1\} }$. Let $\displaystyle{ Y }$ be the received variable with alphabet $\displaystyle{ \{0,1,\text{e} \} }$, where $\displaystyle{ \text{e} }$ is the erasure symbol. Then, the channel is characterized by the conditional probabilities:[1]

\displaystyle{ \begin{align} \operatorname {Pr} [ Y = 0 | X = 0 ] &= 1 - P_e \\ \operatorname {Pr} [ Y = 0 | X = 1 ] &= 0 \\ \operatorname {Pr} [ Y = 1 | X = 0 ] &= 0 \\ \operatorname {Pr} [ Y = 1 | X = 1 ] &= 1 - P_e \\ \operatorname {Pr} [ Y = e | X = 0 ] &= P_e \\ \operatorname {Pr} [ Y = e | X = 1 ] &= P_e \end{align} }

## Capacity

The channel capacity of a BEC is $\displaystyle{ 1-P_e }$, attained with a uniform distribution for $\displaystyle{ X }$ (i.e. half of the inputs should be 0 and half should be 1).[2]

If the sender is notified when a bit is erased, they can repeatedly transmit each bit until it is correctly received, attaining the capacity $\displaystyle{ 1-P_e }$. However, by the noisy-channel coding theorem, the capacity of $\displaystyle{ 1-P_e }$ can be obtained even without such feedback.[3]

## Related channels

If bits are flipped rather than erased, the channel is a binary symmetric channel (BSC), which has capacity $\displaystyle{ 1 - \operatorname H_\text{b}(P_e) }$ (for the binary entropy function $\displaystyle{ \operatorname{H}_\text{b} }$), which is less than the capacity of the BEC for $\displaystyle{ 0\lt P_e\lt 1/2 }$.[4][5] If bits are erased but the receiver is not notified (i.e. does not receive the output $\displaystyle{ e }$) then the channel is a deletion channel, and its capacity is an open problem.[6]

## History

The BEC was introduced by Peter Elias of MIT in 1955 as a toy example.