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 (i.e., transversal single-qubit Clifford gates are realized within a single code block, while transversal two-qubit Clifford gates are realized across two identical code blocks). 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：本文提出了一种构造Calderbank-Shor-Steane(CSS)码的框架，此类码支持容错逻辑横向$Z$-旋转。利用该框架，我们获得了渐近优良的CSS码，能够容错地实现逻辑横向Clifford群（即：横向单量子比特Clifford门在单个码块内实现，而横向双量子比特Clifford门在两个相同码块间实现）。此外，通过研究CSS-T码，我们：(a)证明了存在渐近优良的CSS-T码，其中横向$T$实现逻辑横向$S^{\dagger}$；(b)表明$C_2 \ast C_1 \subseteq C_1^{\perp}$条件是CSS-T码的必要非充分条件；以及(c)修订了CSS-T码的不同刻画，其中横向$T$分别实现逻辑恒等门和逻辑横向$T$门。