A binary symmetric channel (or BSC_p) is a common communications channel model used in coding theory and information theory. In this model, a transmitter wishes to send a bit (a zero or a one), and the receiver will receive a bit. The bit will be "flipped" with a "crossover probability" of p, and otherwise is received correctly. This model can be applied to varied communication channels such as telephone lines or disk drive storage.

The noisy-channel coding theorem applies to BSC_p, saying that information can be transmitted at any rate up to the channel capacity with arbitrarily low error. The channel capacity is ${\displaystyle 1-\operatorname {H} _{\text{b}}(p)}$ bits, where ${\displaystyle \operatorname {H} _{\text{b}}}$ is the binary entropy function. Codes including Forney's code have been designed to transmit information efficiently across the channel.

A binary symmetric channel with crossover probability ${\displaystyle p}$, denoted by BSC_p, is a channel with binary input and binary output and probability of error ${\displaystyle p}$. That is, if ${\displaystyle X}$ is the transmitted random variable and ${\displaystyle Y}$ the received variable, then the channel is characterized by the conditional probabilities:

It is assumed that ${\displaystyle 0\leq p\leq 1/2}$. If ${\displaystyle p>1/2}$, then the receiver can swap the output (interpret 1 when it sees 0, and vice versa) and obtain an equivalent channel with crossover probability ${\displaystyle 1-p\leq 1/2}$.

The channel capacity of the binary symmetric channel, in bits, is:

where ${\displaystyle \operatorname {H} _{\text{b}}(p)}$ is the binary entropy function, defined by:

Shannon's noisy-channel coding theorem gives a result about the rate of information that can be transmitted through a communication channel with arbitrarily low error. We study the particular case of ${\displaystyle {\text{BSC}}_{p}}$.

The noise ${\displaystyle e}$ that characterizes ${\displaystyle {\text{BSC}}_{p}}$ is a random variable consisting of n independent random bits (n is defined below) where each random bit is a ${\displaystyle 1}$ with probability ${\displaystyle p}$ and a ${\displaystyle 0}$ with probability ${\displaystyle 1-p}$. We indicate this by writing "${\displaystyle e\in {\text{BSC}}_{p}}$".