  We study variants of Shor's code that are adept at handling single-axis
correlated idling errors, which are commonly observed in many quantum systems.
By using the repetition code structure of the Shor's code basis states, we
calculate the logical channel applied to the encoded information when subjected
to coherent and correlated single qubit idling errors, followed by stabilizer
measurement. Changing the signs of the stabilizer generators allows us to
change how the coherent errors interfere, leading to a quantum error correcting
code which performs as well as a classical repetition code of equivalent
distance against these errors. We demonstrate a factor of 4 improvement of the
logical memory in a distance-3 logical qubit implemented on a trapped-ion
quantum computer. Even-distance versions of our Shor code variants are
decoherence-free subspaces and fully robust to identical and independent
coherent idling noise.
