Generalized ${\mathbb{Z}}_p$ toric codes as qudit low-density parity-check codes

We study two-dimensional translation-invariant CSS stabilizer codes over prime-dimensional qudits on the square lattice under twisted boundary conditions, generalizing the Kitaev ${\mathbb{Z}}_p$ toric code by augmenting each stabilizer with two additional qudits. Using the Laurent-polynomial formalism, we adapt the Gröbner basis to compute the logical dimension $k$ efficiently, without explicitly constructing large parity-check matrices. We then perform a systematic search over various stabilizer realizations and lattice geometries for $p\in \{3,5,7,11\}$, identifying qudit low-density parity-check codes with the optimal finite-size performance. Representative examples include $[[242,10,22]{]}_3$ and $[[120,6,20]{]}_{11}$, both achieving $k{d}^2\mathrm{/}n=20$. Across the searched regime, the best observed $k{d}^2$ at fixed $n$ increases with $p$, with an empirical relation $k{d}^2=0.0541n^2\ln p+3.84n$, compatible with a Bravyi–Poulin–Terhal-type tradeoff when the interaction range grows with system size.