纠缠辅助量子局部可恢复码：具有可用性的界与构造方法
Entanglement-assisted quantum locally recoverable codes: bounds and constructions with availability
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中文总结 AI 辅助
本研究定义了具有可用性的纠缠辅助量子局部可恢复码，建立了其类Singleton界，基于经典线性码给出随机构造，并从Tamo-Barg等经典码族给出显式构造，证明共享纠缠可支持多个局部恢复集。
中文摘要 AI 辅助
在本研究中，我们定义了具有可用性的纠缠辅助量子局部可恢复码，其中任意最多δ-1个被擦除的量子位可从t个局部恢复集中的任意一个恢复，每个局部恢复集的大小最多为r+δ-1，且恢复集恰好相交于被擦除的坐标，r为一个（较小的）正整数。我们证明共享纠缠允许t>1，即对于同一组最多δ-1个擦除，可提供多个局部恢复集。我们为这类码建立了类Singleton界，并基于具有范德蒙德奇偶校验矩阵的经典线性码给出随机构造。此外，我们还从多个经典码族及其折叠版本，包括Tamo-Barg码、纤维积码以及代数几何码（如单点埃尔米特码和铃木码），给出了具有可用性的纠缠辅助量子局部可恢复码的显式构造。
英文摘要
In this work, we define entanglement-assisted quantum locally recoverable codes with availability, in which any set of up to $δ-1$ erased qudits can be recovered from any one of $t$ local recovery sets, each of size at most $r+δ-1$, with the recovery sets intersecting exactly in the erased coordinates, where $r$ is a (small) positive integer. We show that shared entanglement permits $t>1$, meaning that multiple local recovery sets can be available for the same set of up to $δ-1$ erasures. We establish a Singleton-like bound for this family of codes and present random constructions based on classical linear codes with Vandermonde parity-check matrices. We also provide explicit constructions of entanglement-assisted quantum locally recoverable codes with availability from several classical code families and their folded versions, including Tamo-Barg codes, fiber-product codes, and algebraic-geometry codes such as one-point Hermitian and Suzuki codes.
