Binary Symmetric Channel: capacity and the repetition code

What you are seeing: the Binary Symmetric Channel (BSC) is the simplest noisy channel: a transmitted bit flips with probability $p$, stays with probability $1 - p$. Shannon's noisy-channel theorem gives capacity $C(p) = 1 - H(p)$ where $H(p) = -p \log_2 p - (1-p) \log_2 (1-p)$ is the binary entropy. $C = 0$ at $p = 0.5$ and $C = 1$ at $p = 0$ or $1$.

The top panel plots $C(p)$ and $H(p)$. The bottom panel plots the bit-error-rate of an $n$-repetition code (transmit each bit $n$ times, decode by majority vote) for $n = 1, 3, 5, 7, 11$. Repetition trades rate for reliability: the error drops fast with $n$ at fixed $p \lt 0.5$, at the cost of $n$ channel uses per source bit.