Existence and characterization of additional phantom code classes

Determine whether additional classes of CSS phantom stabilizer codes exist beyond the [[12,2,4]] Carbon code and the [[2^D,D,2]] hypercube family; characterize the structural constraints such codes obey (including stabilizer structure, automorphism groups, and distance trade-offs); and ascertain whether their architectural advantages persist for scalable applications under realistic circuit-level noise.

Background

Phantom codes are stabilizer codes in which every ordered pair of logical qubits admits a logical CNOT implemented purely by qubit permutations, allowing entangling circuits to compile away to zero depth. Prior to this work, only two examples were known: the [[12,2,4]] Carbon code and the [[2^D,D,2]] hypercube family. While the paper greatly expands the catalog and provides construction methods, the broader landscape and limits of phantom codes remain largely unexplored.

A central question is whether substantially different, possibly higher-distance or higher-rate, phantom code families exist, what structural constraints they must satisfy, and whether their practical advantages survive under realistic noise when scaled to large systems.