Optimal Pure Quantum $(r,δ)$-LRCs from Euclidean and Hermitian Dual-Containing Cyclic Codes
Locally recoverable codes (LRCs) combine global error correction with the ability to repair a small number of erased coordinates by accessing only a limited number of other coordinates. Motivated by their quantum counterparts, we construct several families of optimal cyclic $(r,δ)$-LRCs that are Euclidean or Hermitian dual-containing. In the Euclidean case, we obtain four families of optimal dual-containing cyclic $(r,δ)$-LRCs over $\mathbb{F}_q$ with a range of minimum distances extending beyond the local distance $δ$. In the Hermitian case, we derive analogous families over $\mathbb{F}_{q^2}$ that are Hermitian dual-containing, when $(r+δ-1)\mid(q^2-1)$. In addition, we develop a distinct construction for the case $(r+δ-1)\mid(q^2+1)$ using symmetric defining sets and odd $δ$, which yields optimal codes with minimum distances $\ell+2$, $δ+2$, $2δ-2$, and $2δ$. The Euclidean Calderbank-Shor-Steane (CSS) and Hermitian stabilizer constructions then give corresponding quantum cyclic $(r,δ)$-LRCs over~$\mathbb{F}_q$. Further, the resulting stabilizer codes are pure; they meet the relevant quantum Singleton-type bound, and are therefore \textit{optimal}. We also provide a comparison with previous works, highlighting the parameter regimes and minimum distance ranges covered by our results that are not attained by existing constructions. Explicit examples illustrate the constructions and verify the dual-containment and purity conditions.