具有相交恢复集的CSS量子LRC：构造与界
CSS Quantum LRCs with Intersecting Recovery Sets: Constructions and Bounds
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中文总结 AI 辅助
该研究构造了具有相交恢复集的CSS量子本地可恢复码，推导其相关界并证明构造的码族兼具高码率与非平凡最小距离。
中文摘要 AI 辅助
本研究探讨具有局域性r、每个量子位（qudit）对应t个恢复集、相交参数x的(r,t,x)量子本地可恢复码（qLRC）。首先证明，若底层经典码的对偶最小距离至少为2，则CSS码为(r,t,x)-qLRC当且仅当底层经典码为具有公共恢复集的(r,t,x)经典本地可恢复码（cLRC）。随后利用子集包含矩阵构造二元含对偶的(r,t,x)-cLRC族，经CSS构造得到二元(r,t,x)-qLRC。针对CSS (r,t,x)-qLRC，推导其维数与码率的上界、纯情形的最小距离界及精确情形的类Singleton维数界。最终表明这些码族达到高码率与非平凡最小距离。
英文摘要
In this work, we study $(r,t,x)$ quantum locally recoverable codes (qLRCs) with locality $r$, $t$ recovery sets per qudit, and intersection parameter $x$. We first show that, assuming the underlying classical codes have dual minimum distance at least two, a CSS code is an $(r,t,x)$-qLRC if and only if the underlying classical codes are $(r,t,x)$ classical LRCs (cLRCs) with common recovery sets. We then use subset-inclusion matrices to construct families of binary dual-containing $(r,t,x)$-cLRCs, which yield binary $(r,t,x)$-qLRCs via the CSS construction. For CSS $(r,t,x)$-qLRCs, we derive upper bounds on the dimension and rate, minimum-distance bounds in the pure case, and a Singleton-like dimension bound in the exact case. Finally, we show that these families attain high rates and nontrivial minimum distances.
