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It has been shown that $[[15,1,3]]$ quantum Reed-Muller code is the smallest stabilizer code with a transversal T gate. What are the smallest codes with transversal CCZ or CS gates (which are the other examples of level 3 operators in Clifford hierarchy)? Is it $[[8,3,2]]$ color code for CCZ? What about CS?

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The same paper covers the cases of CCZ and CS (see lemma 4). It's the same code that is the smallest distance 3 code. If you want to extend to distance 2 error detecting codes, I don't know what the limits are.

Full disclosure: I'm one of the authors of the linked paper.

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    $\begingroup$ Thanks! I must have missed that. Great paper btw! $\endgroup$
    – tomek
    Dec 13, 2023 at 10:15

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