Concept

# Binary erasure channel

Summary
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 P_e receives a message that the bit was not received ("erased") . Definition A binary erasure channel with erasure probability P_e is a channel with binary input, ternary output, and probability of erasure P_e. That is, let X be the transmitted random variable with alphabet {0,1}. Let Y be the received variable with alphabet {0,1,\text{e} }, where \text{e} is the erasure symbol. Then, the channel is characterized by the conditional probabilities: :\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
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