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arXiv:2606.09522

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

Rajendra Prasad Rajpurohit, Maheshanand Bhaintwal

Published Jun 8, 2026• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.2606.09522

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Optimal Pure Quantum (r,δ)(r,δ)(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)(r,δ)(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)(r,δ)(r,δ)-LRCs over Fq\mathbb{F}_qFq​ with a range of minimum distances extending beyond the local distance δδδ. In the Hermitian case, we derive analogous families over Fq2\mathbb{F}_{q^2}Fq2​ that are Hermitian dual-containing, when (r+δ−1)∣(q2−1)(r+δ-1)\mid(q^2-1)(r+δ−1)∣(q2−1). In addition, we develop a distinct construction for the case (r+δ−1)∣(q2+1)(r+δ-1)\mid(q^2+1)(r+δ−1)∣(q2+1) using symmetric defining sets and odd δδδ, which yields optimal codes with minimum distances ℓ+2\ell+2ℓ+2, δ+2δ+2δ+2, 2δ−22δ-22δ−2, and 2δ2δ2δ. The Euclidean Calderbank-Shor-Steane (CSS) and Hermitian stabilizer constructions then give corresponding quantum cyclic (r,δ)(r,δ)(r,δ)-LRCs over~Fq\mathbb{F}_qFq​. 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.
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Classical locally recoverable codes (LRCs) have become indispensable in distributed storage systems. They provide efficient recovery in terms of localized errors. Quantum LRCs have very recently been introduced for their potential application in quantum data storage. In this paper, we use classical LRCs to investigate quantum LRCs. We prove that the parameters of quantum LRCs are bounded by their classical counterparts. We deduce bounds on the parameters of quantum LRCs from bounds on the parameters of the classical ones. We establish a characterization of optimal pure quantum LRCs based on classical codes with specific properties. Using well-crafted classical LRCs as ingredients in the construction of quantum CSS codes, we offer the first construction of several families of optimal pure quantum LRCs.
Quantum (r,δ)(r,δ)(r,δ)-Locally Recoverable BCH and Homothetic-BCH Codes
Quantum (r,δ)(r,δ)(r,δ)-locally recoverable codes ((r,δ)(r,δ)(r,δ)-LRCs) are the quantum version of classical (r,δ)(r,δ)(r,δ)-LRCs designed to recover multiple failures in large-scale distributed and cloud storage systems. A quantum (r,δ)(r,δ)(r,δ)-LRC, Q(C)Q(C)Q(C), can be constructed from an (r,δ)(r,δ)(r,δ)-LRC, CCC, which is Euclidean or Hermitian dual-containing. This article is devoted to studying how to get quantum (r,δ)(r,δ)(r,δ)-LRCs from BCH and homothetic-BCH codes. As a consequence, we give pure quantum (r,δ)(r,δ)(r,δ)-LRCs which are optimal for the Singleton-like bound.
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