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In this note we show that the weight enumerators of a real quantum errorcorrecting code with $ X $ and $ Z $ exactly transversal must satisfy certainidentities. One consequence of these identities is that if the code is errordetecting then it is automatically error correcting for free; implying arelationship between transversality and code distance.

Quantum Weight Enumerators for Real Codes with $X$ and $Z$ Exactly Transversal