Cyclic ternary channel capacity ($C={\log }_{2}3-h_2(q)$)

A discrete memoryless channel on the additive group $\mathbb{Z}/3\mathbb{Z}$ maps $x$ to $x$ with probability $1-q$ and to $x+1$ with probability $q$. Every input gives the same conditional entropy $h_2(q)$, whereas the output Shannon entropy is at most ${\log }_{2}3$. A uniform input produces a uniform output since every column of the transition matrix sums to one. Hence the channel capacity is ${\log }_{2}3-h_2(q)$ bits per use. This cyclic-error channel differs from a channel distributing its error probability equally over both other output symbols.