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
Incoding 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 receives a message that the bit was not received ("erased") .
A binary erasure channel with erasure probability is a channel with binary input, ternary output, and probability of erasure . That is, let be the transmitted random variable with alphabet . Let be the received variable with alphabet , where is the erasure symbol. Then, the channel is characterized by the conditional probabilities:[1]
as is known from (and equal to) y unless , which has probability .
By definition , so
.
If the sender is notified when a bit is erased, they can repeatedly transmit each bit until it is correctly received, attaining the capacity . However, by the noisy-channel coding theorem, the capacity of can be obtained even without such feedback.[3]
If bits are flipped rather than erased, the channel is a binary symmetric channel (BSC), which has capacity (for the binary entropy function), which is less than the capacity of the BEC for .[4][5] If bits are erased but the receiver is not notified (i.e. does not receive the output ) then the channel is a deletion channel, and its capacity is an open problem.[6]