Binary symmetric channel

A binary symmetric channel independently flips each transmitted bit with a fixed crossover probability $p$.
Its channel matrix is

An error is undetected by codeword-membership checking when the received word is a different valid codeword. For a linear code this occurs exactly when the nonzero channel-error vector is itself a codeword.

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.

If $p<1/4$, choose $p<\delta <1/4$. Binary codes of relative minimum distance above $2\delta$ have positive asymptotic rate by a greedy packing bound, and nearest-codeword decoding fails only when more than $\delta n$ channel errors occur. That probability tends to zero.