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arXiv:2608.21852

Quantum Locally Repairable Codes from Negacyclic and Repeated-Root Cyclic Codes over Small Fields

Ruipan Yang, Qiang Fu, Liangdong Lu

Published Aug 22, 2026• 0 citations
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Abstract

Quantum locally recoverable codes (qLRCs), introduced recently by Golowich and Guruswami, allow any single-qudit erasure to be recovered from a small set of other qudits. Most known constructions require a large alphabet. We systematically investigate qLRCs obtained, via the CSS construction, from classical constacyclic codes over small fields $\Ff_q$ with $q\in\{2,3,4,5,7\}$. First, we prove that a nonzero dual-containing $λ$-constacyclic code exists only when $λ^2=1$, so that negacyclic and (repeated-root) cyclic codes exhaust the constacyclic route to qLRCs. Second, we show that the locality of a constacyclic code equals the minimum distance of its dual minus one, and we give a simple purity criterion for the resulting quantum codes. Third, we show that odd-like duadic codes whose splitting is given by $μ_{-1}$ yield pure qLRCs; specializing to $q$-ary quadratic residue codes of prime length $p\equiv 3 \pmod 4$ gives an infinite family of pure qLRCs with unbounded minimum distance and certified locality. Finally, by means of concrete computations, we obtain a classification of qLRCs from cyclic, negacyclic, and repeated-root cyclic codes of moderate lengths, which contains the first binary qLRCs from repeated-root cyclic codes and many parameter sets that cyclic codes cannot attain.

DOI
10.48550/arxiv.2608.21852

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Quantum locally recoverable codes (qLRCs), introduced recently by Golowich and Guruswami, allow any single-qudit erasure to be recovered from a small set of other qudits. Most known constructions require a large alphabet. We systematically investigate qLRCs obtained, via the CSS construction, from classical constacyclic codes over small fields \Ffq\Ff_q\Ffq​ with q∈{2,3,4,5,7}q\in\{2,3,4,5,7\}q∈{2,3,4,5,7}. First, we prove that a nonzero dual-containing λλλ-constacyclic code exists only when λ2=1λ^2=1λ2=1, so that negacyclic and (repeated-root) cyclic codes exhaust the constacyclic route to qLRCs. Second, we show that the locality of a constacyclic code equals the minimum distance of its dual minus one, and we give a simple purity criterion for the resulting quantum codes. Third, we show that odd-like duadic codes whose splitting is given by μ−1μ_{-1}μ−1​ yield pure qLRCs; specializing to qqq-ary quadratic residue codes of prime length p≡3(mod4)p\equiv 3 \pmod 4p≡3(mod4) gives an infinite family of pure qLRCs with unbounded minimum distance and certified locality. Finally, by means of concrete computations, we obtain a classification of qLRCs from cyclic, negacyclic, and repeated-root cyclic codes of moderate lengths, which contains the first binary qLRCs from repeated-root cyclic codes and many parameter sets that cyclic codes cannot attain.
Constructions of Quantum (r,δ)(r,δ)(r,δ)-LRCs from cyclic codes
Classical (r,δ)(r,δ)(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)(r,δ)(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)(r,δ)(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.
In this work, we study (r,t,x)(r,t,x)(r,t,x) quantum locally recoverable codes (qLRCs) with locality rrr, ttt recovery sets per qudit, and intersection parameter xxx. We first show that, assuming the underlying classical codes have dual minimum distance at least two, a CSS code is an (r,t,x)(r,t,x)(r,t,x)-qLRC if and only if the underlying classical codes are (r,t,x)(r,t,x)(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)(r,t,x)(r,t,x)-cLRCs, which yield binary (r,t,x)(r,t,x)(r,t,x)-qLRCs via the CSS construction. For CSS (r,t,x)(r,t,x)(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.
Locally recoverable codes (LRCs) with locality parameter r can recover any erased code symbol by accessing r other code symbols. This local recovery property is of great interest in large-scale distributed classical data storage systems as it leads to efficient repair of failed nodes. A well-known class of optimal (classical) LRCs are subcodes of Reed-Solomon codes constructed using a special type of polynomials called good polynomials. Recently, Golowich and Guruswami initiated the study of quantum LRCs (qLRCs), which could have applications in quantum data storage systems of the future. The authors presented a qLRC construction based on good polynomials arising out of subgroups of the multiplicative group of finite fields. In this paper, we present a qLRC construction method that can employ any good polynomial. We also propose a new approach for designing good polynomials using subgroups of affine general linear groups. Golowich and Guruswami also derived a lower bound on the minimum distance of their qLRC under the restriction that r+1 is prime. Using similar techniques in conjunction with the expander mixing lemma, we develop minimum distance lower bounds for our qLRCs without the r+1 prime restriction.
Quantum locally recoverable codes (QLRCs) have recently gained attention as a framework for achieving efficient quantum storage with local recovery capabilities. Analogous to their classical counterparts, QLRCs allow a lost qudit to be reconstructed using only a small subset of other qudits, thereby reducing the resource and operational overhead in recovery. In this work, we extend the study of QLRCs by considering (r,δ)(r,δ)(r,δ) QLRCs characterized by locality parameter rrr and local distance δ≥2δ\geq 2δ≥2. We present constructions of both random and explicit (r,δ)(r,δ)(r,δ) QLRCs, including explicit families based on the quantum Tamo--Barg construction. We also present an efficient decoding algorithm for these quantum Tamo--Barg codes. Furthermore, we introduce quantum \emph{hierarchical} locally recoverable codes (QHLRCs), which extend local recovery to multiple hierarchical levels. For any integer h≥2h\geq 2h≥2, we construct both random and explicit hhh-level QHLRCs, the latter being hhh-level quantum Tamo--Barg codes, and establish a Singleton-like bound for these codes using a CSS framework built from dual-containing classical codes. These results advance the theoretical foundations of quantum erasure recovery and contribute to the design of efficient quantum storage architectures.
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.
Alphabet-Dependent Bounds for Pure Quantum (r,ρ)(r,ρ)(r,ρ)-Locally Recoverable Codes
A quantum (r,ρ)(r,ρ)(r,ρ)-locally recoverable code ((r,ρ)(r,ρ)(r,ρ)-qLRC) is a quantum code in which every qudit can be recovered from at most r+ρ−1r+ρ-1r+ρ−1 other qudits, even after ρ−1ρ-1ρ−1 additional erasures inside the recovery set. The bounds currently known for this class, namely the Singleton-like and the GG Singleton-like bounds, are alphabet independent and are therefore loose for small-to-moderate qudit dimensions. In this letter, we derive three alphabet-dependent upper bounds for pure (r,ρ)(r,ρ)(r,ρ)-qLRCs obtained through the Hermitian CSS construction: a Griesmer-like, a Plotkin-like, and a sphere-packing-like bound. We further establish the asymptotic hierarchy among these bounds and identify the relative-distance regions in which each of them yields the tightest rate constraint.