Bounds for Pure Disjoint $(r,\delta)$-Quantum Locally Recoverable Codes

Abstract

We study pure disjoint $(r,\delta)$-quantum locally recoverable codes (qLRCs) without assuming a stabilizer structure. We formulate local Knill--Laflamme conditions for recovery from up to $\delta-1$ erasures within a recovery block, and introduce blockwise Shor--Laflamme and unitary weight enumerators that capture how error weight is distributed across recovery sets. We establish several properties of these enumerators and use them to derive a Singleton-like bound that strengthens the known bound for disjoint $(r,\delta)$-qLRCs under a purity assumption, as well as a linear-programming upper bound on the code dimension. These results provide a non-stabilizer, weight-enumerator-based approach to the study of pure disjoint $(r,\delta)$-qLRCs.