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The Shannon second coding theorem states that the operational channel capacity of a finite discrete memoryless channel is $C={\max }_{p(x)}I(X:Y)$: rates below this maximum admit block codes with error tending to zero, and rates above it cannot have vanishing error. For this channel the preceding calculation givesusing maximum entropy on a finite alphabet. The transition matrix is a doubly stochastic matrix. Therefore the uniform input has uniform output: $p(y)=\frac{1}{3}{\sum }_{x}p(y\mid x)=1/3$. It achieves the entropy upper bound, and henceEquivalently this is the weakly symmetric channel capacity theorem: permutations of a common row and equal column sums make the uniform input optimal. If $q$ is uniform, the output contains no information about the input and the formula gives zero.

Every row of the channel matrix is a permutation of $(1-2\alpha ,\alpha ,\alpha )$, and every column sum is one. It is therefore a weakly symmetric channel. The uniform input makes the output uniform, with entropy ${\log }_{2}3$, while the conditional output entropy is the entropy of one row:The weakly symmetric channel capacity formula now gives the ternary symmetric channel capacitybits per channel use, for $0\le \alpha \le 1/2$, with the convention $0{\log }_{2}0=0$.