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

2026-08-11

AI Summary

This work investigates the erasure recovery capability and theoretical bounds of pure non-stabilizer $(r,\delta)$-quantum locally recoverable codes (qLRCs) under up to $\delta-1$ erasures. By establishing a local Knill–Laflamme condition, introducing block Shor–Laflamme and unitary weight enumerators, and characterizing the distribution of error weights over recovery sets, the authors combine these tools with linear programming techniques to derive, for the first time in a non-stabilizer setting, a new Singleton-type bound and a tight upper bound on the code dimension for pure qLRCs. These results significantly strengthen existing theoretical limits and deepen the understanding of the structure and performance of non-stabilizer quantum LRCs.

Abstract

We study pure disjoint $(r,δ)$-quantum locally recoverable codes (qLRCs) without assuming a stabilizer structure. We formulate local Knill--Laflamme conditions for recovery from up to $δ-1$ 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 $(r,δ)$-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 $(r,δ)$-qLRCs.

Problem

• quantum locally recoverable codes
• pure codes
• disjoint recovery sets
• Singleton bound
• weight enumerators

Innovation

• quantum locally recoverable codes
• weight enumerators
• Knill-Laflamme conditions
• Singleton bound
• non-stabilizer codes