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At crossover probability $p=1/2$, the two rows of the transition matrix of a binary symmetric channel coincide. Its output is then independent of its input and its capacity is zero.

A reliable transmission rate $R$ means that length-$n$ codes can carry $M_n$ messages with $n^{-1}{\log }_2{M}_n$ tending to at least $R$ while the probability measure of the decoding-error event tends to zero. By the Shannon second coding theorem, the supremum of reliable rates is the binary symmetric channel capacityFor small positive $p$, the binary entropy function has the asymptotic expansionso $C(p)$ starts at one bit per channel use and decreases rapidly from one as errors are introduced.

A binary symmetric channel independently flips each input bit with probability $p$, so, with rows indexed by the input and columns by the output, its channel matrix isIf $p>1/2$, complementing every received bit converts the channel into one with crossover probability $1-p<1/2$ without changing its information-carrying ability. The case $p=1/2$ has identical rows and capacity zero, so it suffices to study $p<1/2$.

The Shannon second coding theorem states that for every rate below the channel capacity there are arbitrarily long codes whose decoding error tends to zero, whereas no sequence of codes with rate above capacity can have vanishing error. It also identifiesFor the present channel,The output entropy is at most one bit, with equality for a uniform input. Hence the binary symmetric channel capacity isbits per channel use.