Classical $(r,δ)$ locally recoverable codes (LRCs) play a central role in distributed data storage systems as they enable an efficient recovery from erasures by accessing a small number of surviving symbols. Motivated by their prospective use in future quantum data storage and by recent theoretical progress on quantum locally recoverable codes (qLRCs), we investigate the construction of qLRCs from classical cyclic $(r,δ)$-LRCs. Our approach identifies cyclic LRCs whose defining sets satisfy a dual-containing condition, allowing them to serve as valid CSS ingredients. We present three explicit families of $(r,δ)$-qLRCs, two of which are optimal with respect to the quantum Singleton-like bound, whenever the codes are pure, thereby providing optimal examples. Additionally, the codes presented in Constructions 2 and 3 have no bound on their lengths with respect to the field size required to obtain these codes.

翻译：经典$(r,δ)$-局部可修复码（LRCs）在分布式数据存储系统中发挥着核心作用，因其可通过访问少量幸存符号实现擦除的高效修复。受其在未来量子数据存储中的应用前景以及量子局部可修复码（qLRCs）理论最新进展的启发，我们研究了从经典循环$(r,δ)$-LRCs构造qLRCs的方法。本方法识别出定义集满足对偶包含条件的循环LRCs，使其可作为有效的CSS构造要素。我们提出了三个显式的$(r,δ)$-qLRCs族系，其中两个族系在纯码情形下相对于量子单子上界达到最优，从而提供了最优实例。此外，构造2和构造3给出的码在所需域大小方面对码长无约束。