Asymptotically good CSS-T codes and a new construction of triorthogonal codes

We propose a new systematic construction of CSS-T codes from any given CSS code using a map $\phi$. When $\phi$ is the identity map $I$, we retrieve the construction of [1] and use it to prove the existence of asymptotically good binary CSS-T codes, resolving a previously open problem in the literature, and of asymptotically good quantum LDPC CSS-T codes. We analyze the structure of the logical operators corresponding to certain non-Clifford gates supported by the quantum codes obtained from this construction ($\phi =I$), concluding that they always result in the logical identity. An immediate application of these codes in dealing with coherent noise is discussed. We then develop a new doubling transformation for obtaining triorthogonal codes, which generalizes the doubling construction presented in [2]. Our approach permits using self-orthogonal codes, instead of only doubly-even codes, as building blocks for triorthogonal codes. This broadens the range of codes available for magic state distillation.