Toward Quantum CSS-T Codes from Sparse Matrices

CSS-T codes were recently introduced as quantum error-correcting codes that respect a transversal gate. A CSS-T code depends on a pair (C₁, C₂) of binary linear codes C₁ and C₂ that satisfy certain conditions. We prove that C₁ and C₂ form a CSS-T pair if and only if C₂ Hull (C₁) Hull (C₁²), where the hull of a code is the intersection of the code with its dual. We show that if (C₁, C₂) is a CSS-T pair, and the code C₂ is degenerated on \i\, meaning that the i^th-entry is zero for all the elements in C₂, then the pair of punctured codes (C₁|ᵢ, C₂|ᵢ) is also a CSS-T pair. Finally, we provide Magma code based on our results and quasi-cyclic codes as a step toward finding quantum LDPC or LDGM CSS-T codes computationally.