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
$C_{2}\subset \text{Hull}(C_{1})\cap \text{Hull}(C_{1}^{2})$
, 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 (
$C_{1}\vert_{i}, C_{2}\vert_{i}$
) 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.