Alphabet-Dependent Bounds for Pure Quantum (r,ρ)-Locally Recoverable Codes

A quantum (r,ρ)-locally recoverable code ((r,ρ)-qLRC) is a quantum code in which every qudit can be recovered from at most r+ρ-1 other qudits, even after ρ-1 additional erasures inside the recovery set. The bounds currently known for this class, namely the Singleton-like and the GG Singleton-like bounds, are alphabet independent and are therefore loose for small-to-moderate qudit dimensions. In this letter, we derive three alphabet-dependent upper bounds for pure (r,ρ)-qLRCs obtained through the Hermitian CSS construction: a Griesmer-like, a Plotkin-like, and a sphere-packing-like bound. We further establish the asymptotic hierarchy among these bounds and identify the relative-distance regions in which each of them yields the tightest rate constraint.