CSS-T codes were recently introduced as quantum error-correcting codes that respect a transversal gate. A CSS-T code depends on a CSS-T pair, which is a pair of binary codes \((C_1, C_2)\) such that \(C_1\) contains \(C_2\) , \(C_2\) is even, and the shortening of the dual of \(C_1\) with respect to the support of each codeword of \(C_2\) is self-dual. In this paper, we give new conditions to guarantee that a pair of binary codes \((C_1, C_2)\) is a CSS-T pair. We define the poset of CSS-T pairs and determine the minimal and maximal elements of the poset. We provide a propagation rule for nondegenerate CSS-T codes. We apply some main results to Reed–Muller, cyclic and extended cyclic codes. We characterize CSS-T pairs of cyclic codes in terms of the defining cyclotomic cosets. We find cyclic and extended cyclic codes to obtain quantum codes with better parameters than those in the literature.
