 Logical failure of the three-qubit phase-flip repetition code

The phase-flip repetition code with codewords $|+++⟩,|---⟩$ corrects any one physical Pauli Z gate error. Its two adjacent-pair error syndromes cannot distinguish an error pattern from its three-bit complement. Minimum-weight recovery therefore leaves no residual error at weight zero or one, and leaves $Z_1{Z}_2{Z}_3$ at weight two or three. This residual operator exchanges the codewords, so inverse encoding gives a logical Pauli X gate. For independent physical phase errors of probability $ϵ$, the decoded quantum channel is $(1-p_L)\rho +p_{L}X\rho X$, where $p_L=3{ϵ}^2(1-ϵ)+{ϵ}^3$. For $0<ϵ<1/2$, $ϵ-p_L=ϵ(1-ϵ)(1-2ϵ)>0$.