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For every integer $r\geq 2$ and every $\epsilon>0$, we construct an explicit infinite family of quantum LDPC codes supporting a transversal $C^{r-1}Z$ gate with length $N$, dimension $K\geq N^{1-\epsilon}$, distance $D\geq N^{1/r}/\operatorname{poly}(\log N)$, and stabilizer weight $w\leq\operatorname{poly}(\log N)$.
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