Toward Quantum CSS-T Codes from Sparse Matrices
<|reference_start|>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 $(C_1, C_2)$ of binary linear codes $C_1$ and $C_2$ that satisfy certain conditions. We prove that $C_1$ and $C_2$ form a CSS-T pair if and only if $C_2 \subset \operatorname{Hull}(C_1) \cap \operatorname{Hull}(C_1^2)$, where the hull of a code is the intersection of the code with its dual. We show that if $(C_1,C_2)$ is a CSS-T pair, and the code $C_2$ is degenerated on $\{i\}$, meaning that the $i^{th}$-entry is zero for all the elements in $C_2$, then the pair of punctured codes $(C_1|_i,C_2|_i)$ 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.<|reference_end|>
@article{camps-moreno2024toward,
    title={Toward Quantum CSS-T Codes from Sparse Matrices},
    author={Eduardo Camps-Moreno, Hiram H. L'opez, Gretchen L. Matthews, Emily
  McMillon},
    journal={arXiv preprint arXiv:2406.00425},
    year={2024},
    archivePrefix={arXiv},
    eprint={2406.00425},
    primaryClass={cs.IT math.IT}
}