Time-reversal-invariant logical qubits on an odd number of spins have automatically scalar even-weight Pauli errors, so single-qubit error detection implies correction and every such code has odd distance.
Quantum Weight Enumerators for Real Codes with $X$ and $Z$ Exactly Transversal
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abstract
In this note we show that the weight enumerators of a real quantum error correcting code with $ X $ and $ Z $ exactly transversal must satisfy certain identities. One consequence of these identities is that if the code is error detecting then it is automatically error correcting for free; implying a relationship between transversality and code distance.
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Time-Reversal Selection Rules for Quantum Error Correction
Time-reversal-invariant logical qubits on an odd number of spins have automatically scalar even-weight Pauli errors, so single-qubit error detection implies correction and every such code has odd distance.