CSS Quantum LRCs with Intersecting Recovery Sets: Constructions and Bounds

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.