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.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.

cs.ITCross-listedarXiv:2608.10912Aug 12, 20260 citations

Optimal Pure Quantum ( r , δ )-Locally Recoverable Codes From Matrix-Product Construction

Carlos Galindo, Fernando Hernando, Carlos Munuera +1 more

IEEE Transactions on Information TheoryAug 28, 20260 citations

Entanglement-assisted quantum locally recoverable codes: bounds and constructions with availability

Rutuja Kshirsagar, Gretchen L. Matthews, Julia Shapiro

In this work, we define entanglement-assisted quantum locally recoverable codes with availability, in which any set of up to $δ-1$ erased qudits can be recovered from any one of $t$ local recovery sets, each of size at most $r+δ-1$ , with the recovery sets intersecting exactly in the erased coordinates, where $r$ is a (small) positive integer. We show that shared entanglement permits $t>1$ , meaning that multiple local recovery sets can be available for the same set of up to $δ-1$ erasures. We establish a Singleton-like bound for this family of codes and present random constructions based on classical linear codes with Vandermonde parity-check matrices. We also provide explicit constructions of entanglement-assisted quantum locally recoverable codes with availability from several classical code families and their folded versions, including Tamo-Barg codes, fiber-product codes, and algebraic-geometry codes such as one-point Hermitian and Suzuki codes.In this work, we define entanglement-assisted quantum locally recoverable codes with availability, in which any set of up to $δ-1$ erased qudits can be recovered from any one of $t$ local recovery sets, each of size at most $r+δ-1$, with the recovery sets intersecting exactly in the erased coordinates, where $r$ is a (small) positive integer. We show that shared entanglement permits $t>1$, meaning that multiple local recovery sets can be available for the same set of up to $δ-1$ erasures. We establish a Singleton-like bound for this family of codes and present random constructions based on classical linear codes with Vandermonde parity-check matrices. We also provide explicit constructions of entanglement-assisted quantum locally recoverable codes with availability from several classical code families and their folded versions, including Tamo-Barg codes, fiber-product codes, and algebraic-geometry codes such as one-point Hermitian and Suzuki codes.

cs.ITCross-listedarXiv:2608.09886Aug 11, 20260 citations

Quantum locally recoverable code with intersecting recovery sets

Kaifeng Bu, Weichen Gu, Xiang Li

We introduce the concept of quantum locally recoverable codes (qLRCs) with intersecting recovery sets. We derive a singleton-like bound for these codes by leveraging the additional information provided by the intersecting recovery sets. Furthermore, we provide a construction for qLRCs with intersecting recovery sets by introducing a variation of the hypergraph product. Finally, we apply our qLRC methods to obtain improved results for classical LRCs. These results may provide new insights into the locality of quantum error correction code.We introduce the concept of quantum locally recoverable codes (qLRCs) with intersecting recovery sets. We derive a singleton-like bound for these codes by leveraging the additional information provided by the intersecting recovery sets. Furthermore, we provide a construction for qLRCs with intersecting recovery sets by introducing a variation of the hypergraph product. Finally, we apply our qLRC methods to obtain improved results for classical LRCs. These results may provide new insights into the locality of quantum error correction code.

arXiv:2501.10354Jan 17, 20250 citations

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.

In this work, we define entanglement-assisted quantum locally recoverable codes with availability, in which any set of up to

δ-1

erased qudits can be recovered from any one of

t

local recovery sets, each of size at most

r+δ-1

, with the recovery sets intersecting exactly in the erased coordinates, where

r

is a (small) positive integer. We show that shared entanglement permits

t>1

, meaning that multiple local recovery sets can be available for the same set of up to

δ-1

erasures. We establish a Singleton-like bound for this family of codes and present random constructions based on classical linear codes with Vandermonde parity-check matrices. We also provide explicit constructions of entanglement-assisted quantum locally recoverable codes with availability from several classical code families and their folded versions, including Tamo-Barg codes, fiber-product codes, and algebraic-geometry codes such as one-point Hermitian and Suzuki codes.