Quantum locally recoverable codes (qLRCs) allow a single qudit erasure to be corrected by accessing only a small number of other qudits. Standard CSS and Hermitian constructions, however, impose dual-containing or self-orthogonal constraints on the underlying classical codes, thereby restricting the well-structured classical LRCs (cLRCs) that can be used to construct qLRCs. To relax these constraints, we introduce entanglement-assisted quantum locally recoverable codes (EAQLRCs) by assuming that halves of the pre-shared maximally entangled pairs are noiseless. We characterize sufficient support conditions on extended stabilizers under which entanglement-assisted stabilizer codes have locality $r$ and derive a CSS-like construction from two classical codes without imposing the ordinary dual-containing condition. We further establish an upper bound on locality and a Singleton-like bound for arbitrary CSS-like EAQLRCs, and characterize the pure codes attaining equality in the latter bound. These results yield a general framework for constructing optimal pure EAQLRCs from pairs of cLRCs. Applying this framework to $\ell$ -intersection pairs of MDS codes and block parity-check matrices, we obtain two families of optimal pure CSS-like EAQLRCs with flexible parameters and nontrivial localities. To the best of our knowledge, these represent the first explicit families of EAQLRCs.Quantum locally recoverable codes (qLRCs) allow a single qudit erasure to be corrected by accessing only a small number of other qudits. Standard CSS and Hermitian constructions, however, impose dual-containing or self-orthogonal constraints on the underlying classical codes, thereby restricting the well-structured classical LRCs (cLRCs) that can be used to construct qLRCs. To relax these constraints, we introduce entanglement-assisted quantum locally recoverable codes (EAQLRCs) by assuming that halves of the pre-shared maximally entangled pairs are noiseless. We characterize sufficient support conditions on extended stabilizers under which entanglement-assisted stabilizer codes have locality $r$ and derive a CSS-like construction from two classical codes without imposing the ordinary dual-containing condition. We further establish an upper bound on locality and a Singleton-like bound for arbitrary CSS-like EAQLRCs, and characterize the pure codes attaining equality in the latter bound. These results yield a general framework for constructing optimal pure EAQLRCs from pairs of cLRCs. Applying this framework to $\ell$-intersection pairs of MDS codes and block parity-check matrices, we obtain two families of optimal pure CSS-like EAQLRCs with flexible parameters and nontrivial localities. To the best of our knowledge, these represent the first explicit families of EAQLRCs.

arXiv:2607.27091Jul 29, 20260 citations

Constructions of Quantum $(r,δ)$ -LRCs from cyclic codesConstructions of Quantum $(r,δ)$-LRCs from cyclic codes

Rajendra Prasad Rajpurohit, Maheshanand Bhaintwal

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.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.

arXiv:2606.09522Jun 8, 20260 citations

On Optimal Quantum LRCs From the Hermitian Construction and t -Designs

Yang Li, Shitao Li, Huimin Lao +2 more

In a recent work, quantum locally recoverable codes (qLRCs) have been introduced for their potential application in large-scale quantum data storage and implication for quantum LDPC codes. This work focuses on the bounds and constructions of qLRCs derived from the Hermitian construction, which solves an open problem proposed by Luo $et~al.$ (IEEE Trans. Inf. Theory, 71 (3): 1794-1802, 2025). We present four bounds for qLRCs and give comparisons in terms of their asymptotic formulas. We construct several new infinite families of NMDS codes, with general and flexible dimensions, that support t-designs for $t\in \{2,3\}$ , and apply them to obtain Hermitian dual-containing classical LRCs (cLRCs). As a result, we derive three explicit families of optimal qLRCs. Compared to the known qLRCs obtained by the CSS construction, our optimal qLRCs offer new and more flexible parameters. It is also worth noting that the constructed cLRCs themselves are interesting as they are optimal with respect to four distinct bounds for cLRCs.In a recent work, quantum locally recoverable codes (qLRCs) have been introduced for their potential application in large-scale quantum data storage and implication for quantum LDPC codes. This work focuses on the bounds and constructions of qLRCs derived from the Hermitian construction, which solves an open problem proposed by Luo $et~al.$ (IEEE Trans. Inf. Theory, 71 (3): 1794-1802, 2025). We present four bounds for qLRCs and give comparisons in terms of their asymptotic formulas. We construct several new infinite families of NMDS codes, with general and flexible dimensions, that support t-designs for $t\in \{2,3\}$, and apply them to obtain Hermitian dual-containing classical LRCs (cLRCs). As a result, we derive three explicit families of optimal qLRCs. Compared to the known qLRCs obtained by the CSS construction, our optimal qLRCs offer new and more flexible parameters. It is also worth noting that the constructed cLRCs themselves are interesting as they are optimal with respect to four distinct bounds for cLRCs.

arXiv:2508.13553IEEE Transactions on Information TheoryMay 26, 20263 citations

r

and derive a CSS-like construction from two classical codes without imposing the ordinary dual-containing condition. We further establish an upper bound on locality and a Singleton-like bound for arbitrary CSS-like EAQLRCs, and characterize the pure codes attaining equality in the latter bound. These results yield a general framework for constructing optimal pure EAQLRCs from pairs of cLRCs. Applying this framework to

\ell

-intersection pairs of MDS codes and block parity-check matrices, we obtain two families of optimal pure CSS-like EAQLRCs with flexible parameters and nontrivial localities. To the best of our knowledge, these represent the first explicit families of EAQLRCs.

Constructions of Quantum

(r,δ)

-LRCs from cyclic codes

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.

et~al.

(IEEE Trans. Inf. Theory, 71 (3): 1794-1802, 2025). We present four bounds for qLRCs and give comparisons in terms of their asymptotic formulas. We construct several new infinite families of NMDS codes, with general and flexible dimensions, that support t-designs for

t\in \{2,3\}

, and apply them to obtain Hermitian dual-containing classical LRCs (cLRCs). As a result, we derive three explicit families of optimal qLRCs. Compared to the known qLRCs obtained by the CSS construction, our optimal qLRCs offer new and more flexible parameters. It is also worth noting that the constructed cLRCs themselves are interesting as they are optimal with respect to four distinct bounds for cLRCs.