Phase-flip repetition code

The $n$-qubit phase-flip repetition code is stabilized by $X_j{X}_{j+1}$ and has codewords $|+{⟩}^{\otimes n}$ and $|-{⟩}^{\otimes n}$. It has logical operators $\overline{Z}=X_1$ and $\overline{X}=Z_1\cdots Z_n$, so its distance against phase flips is $n$.

For independent errors of probability $p<1/2$, maximum-likelihood decoding of a repetition code chooses the lower-weight error compatible with the syndrome. This is majority voting, with threshold $p_c=1/2$ and exponentially vanishing logical-error probability below threshold.