Subsystem modular-qudit CSS code

Symplectic doubling: Any \([[n,k,r,d]]_{\mathbb{Z}_q}\) subsystem stabilizer code can be mapped onto a \([[2n,2k,2r,\geq d]]_{\mathbb{Z}_q}\) subsystem CSS code, with the mapping preserving geometric locality of a code up to a constant factor [2] (see also [3][4; Thm. 1]). In the modular symplectic representation, the gauge-group generator matrix of the former is mapped into that of latter as follows, \begin{align} \begin{pmatrix}G_{X} & G_{Z}\end{pmatrix} \to \begin{pmatrix} 0 & 0 & G_{Z} & -G_{X}\\ G_{X} & G_{Z} & 0 & 0 \end{pmatrix}~, \tag*{(2)}\end{align} where the first two columns of the latter matrix correspond to the \(X\)-type part of the gauge-group generator matrix of the output subsystem CSS code. In the case of a stabilizer code, the stabilizer generator matrix is mapped instead to yield a two-block CSS code (see [4; Thm. 1] for the case of qubit stabilizer codes). For geometrically local 2D stabilizer codes with twist defects, this mapping yields a twisted double cover of the underlying qudit geometry [5].