Singleton-Optimal Rank-Metric CSS Codes : Equality Structure and Exact Projected-Recovery Radii
Correlated faults from shared control can have dense physical support yet low rank over a base field, motivating rank-metric quantum codes for stacked architectures. We prove an asymmetric rank-metric Singleton bound for Calderbank--Shor--Steane codes, including degenerate codes, and characterize equality through commuting maximum-rank-distance matrix codes. Optimal pairs exist for every admissible parameter triple and are necessarily pure. Comparison with erasure bounds establishes an exact advantage in stacked rank distance for general stabilizer codes at fixed physical resources on certain tall layouts. We determine whether relaxing the recovery target enlarges the worst-case correctable radius. For Singleton-optimal pairs with positive logical dimension, the measured syndrome determines the projected error modulo stabilizers exactly at radii strictly below half the sector rank distance, on every layout and for every nonzero projector. The projected syndrome is recoverable at arbitrary radii exactly when the complementary check space is invariant under the adjoint projector; otherwise, the same radius limit applies. For nontrivial idempotents, this invariance requires more rows than columns or sector distance one. For trace-self-adjoint projectors, linear ambient projected-syndrome interfaces in both sectors exist exactly when the code splits as a tensor product across the projector. Designs placing the two check spaces in complementary projector images are necessarily one-sided whenever they encode logical information. Two explicit families realize the extreme cases of the equality structure, including a two-sided family with an efficient certifying decoder attaining the optimal unique-decoding radius.