This paper introduces a framework for constructing Calderbank-Shor-Steane (CSS) codes that support fault-tolerant logical transversal $Z$-rotations. Using this framework, we obtain asymptotically good CSS codes that fault-tolerantly realize the logical transversal Clifford group. Furthermore, investigating CSS-T codes, we: (a) demonstrate asymptotically good CSS-T codes wherein the transversal $T$ realizes the logical transversal $S^{\dagger}$; (b) show that the condition $C_2 \ast C_1 \subseteq C_1^{\perp}$ is necessary but not sufficient for CSS-T codes; and (c) revise the characterizations of CSS-T codes wherein the transversal $T$ implements the logical identity and the logical transversal $T$, respectively.

翻译：本文提出了一种用于构造支持容错逻辑横向$Z$旋转的Calderbank-Shor-Steane (CSS)码的框架。利用该框架，我们获得了能够容错实现逻辑横向Clifford群的渐近优CSS码。此外，通过对CSS-T码的研究，我们：(a) 证明了存在渐近优CSS-T码，其横向$T$门可实现逻辑横向$S^{\dagger}$门；(b) 指出条件$C_2 \ast C_1 \subseteq C_1^{\perp}$对于CSS-T码是必要但不充分的；(c) 修正了横向$T$门分别实现逻辑恒等门与逻辑横向$T$门的CSS-T码的特征描述。