Logical Hadamard convention for a phase-flip repetition code

The codewords $|+++⟩,|---⟩$ have residual logical Pauli X gate after an uncorrectable two- or three-site phase error. Precede their encoding by a logical Hadamard gate and follow decoding by its inverse. Since $HXH=Z$, the same error now acts as a logical Pauli Z gate. Explicitly the encoded coefficients are $(\alpha +\beta )/\sqrt{2}$ and $(\alpha -\beta )/\sqrt{2}$ for an input $\alpha |0⟩+\beta |1⟩$. This convention matters when translating physical phase errors into a claimed decoded phase-error channel.