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arXiv:2608.28650

Alphabet-Dependent Bounds for Pure Quantum $(r,ρ)$-Locally Recoverable Codes

Vijay Kumar, Ramakrishna Bandi

quant-phcs.ITSubmitted Aug 19, 2026• Announced Aug 20, 2026• 0 citations
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Abstract

A quantum $(r,ρ)$-locally recoverable code ($(r,ρ)$-qLRC) is a quantum code in which every qudit can be recovered from at most $r+ρ-1$ other qudits, even after $ρ-1$ 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 $(r,ρ)$-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.

Related papers

Entanglement-Assisted Quantum Locally Recoverable Codes: Bounds, Optimal Constructions, and Achievability

Vijay Kumar, Ramakrishna Bandi

This paper studies entanglement-assisted quantum locally recoverable codes (EA-qLRCs) built via a CSS-like stabilizer construction from pairs of classical locally recoverable codes (cLRCs), without requiring dual-containment. We define such codes through local recovery channels, give a sufficient stabilizer criterion for the construction, and derive Singleton-, Griesmer-, Plotkin-, and sphere-packing-like converse bounds on the parameters of the resulting pure CSS-like EA-qLRCs, along with a Cadambe--Mazumdar-like bound that, as in the classical case, lacks a closed form, plus a comparison of their relative tightness across finite-length and asymptotic regimes. We give necessary and sufficient conditions for a pure CSS-like EA-qLRC to attain the Singleton-like bound with equality; for the single-code case $\mathcal{C}_1=\mathcal{C}_2=\mathcal{C}$, this reduces to a simple condition on the hull dimension $s=\dim(\mathcal{C}\cap\mathcal{C}^\perp)$, which also fixes the entanglement count via $c=n-k-s$. We present CSS-like EA-qLRC constructions from classical LRC families---Tamo--Barg and cyclic codes---and characterize when these attain the Singleton-like bound, showing the cyclic families yield optimal codes while the Tamo--Barg construction, though valid, attains the bound only in the degenerate regime $k \le r$, where locality is vacuous. We complement these constructions with two Gilbert--Varshamov-like achievability bounds, via a classical parity-check augmentation and a sharper concatenated-code construction, and show both hold unconditionally for field size $q>3$ via a monomial-equivalence argument. Finally, we unify all bounds---converse and achievability alike---under a common maximally entangled regime, giving a single comparison of the achievable and forbidden rate--distance--locality region for CSS-like EA-qLRCs.

arXiv:2608.06854•Aug 7, 2026•0 citations

CSS Quantum LRCs with Intersecting Recovery Sets: Constructions and Bounds

Evagoras Stylianou, Vinayak Ramkumar, Holger Boche +1 more

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:2608.10912•Aug 12, 2026•0 citations

Asymmetric Entanglement-Assisted Quantum Locally Recoverable Codes

Vijay Kumar, Ramakrishna Bandi

Locally recoverable codes (LRCs) permit reconstruction of a lost coordinate from a small subset of the remaining coordinates, a property that is a fundamental coding technique for distributed storage and, more recently, for fault-tolerant quantum memories and distributed storage systems. Existing constructions of quantum locally recoverable codes (qLRCs) and their entanglement-assisted generalizations (EA-qLRCs) impose a single locality parameter $r$ governing recovery from both bit-flip ($X$-type) and phase-flip ($Z$-type) errors, even though physical noise is rarely symmetric between the two. We introduce entanglement-assisted quantum locally recoverable codes (EA-qLRC) with \emph{asymmetric locality} $\{r_X,r_Z\}$, in which $X$-type and $Z$-type errors are corrected from recovery sets of size at most $r_X+1$ and $r_Z+1$, respectively. We extend a stabilizer sufficient condition for this property and, in particular, the CSS framework, where asymmetric locality corresponds to separate locality conditions on the two classical codes $\mathcal C_1$ and $\mathcal C_2$ underlying the construction. From this characterization we derive Singleton-, Griesmer-, Plotkin-, sphere-packing-, and Cadambe\textendash Mazumdar-like bounds for CSS-type asymmetric EA-qLRCs (AEA-qLRCs) built from Proposition~\ref{CSS-EA-qLRC-Asym}, and we characterize exactly when a pure CSS-type AEA-qLRC built from a pair of classical LRCs attains the Singleton-like bound with equality.

quant-pharXiv:2610.23889•Sep 29, 2026•0 citations

Bounds for Pure Disjoint $(r,δ)$-Quantum Locally Recoverable Codes

Evagoras Stylianou, Holger Boche

We study pure disjoint $(r,δ)$-quantum locally recoverable codes (qLRCs) without assuming a stabilizer structure. We formulate local Knill--Laflamme conditions for recovery from up to $δ-1$ erasures within a recovery block, and introduce blockwise Shor--Laflamme and unitary weight enumerators that capture how error weight is distributed across recovery sets. We establish several properties of these enumerators and use them to derive a Singleton-like bound that strengthens the known bound for disjoint $(r,δ)$-qLRCs under a purity assumption, as well as a linear-programming upper bound on the code dimension. These results provide a non-stabilizer, weight-enumerator-based approach to the study of pure disjoint $(r,δ)$-qLRCs.

cs.ITCross-listedarXiv:2608.10922•IEEE International Symposium on Information Theory (ISIT)•Aug 12, 2026•0 citations

Entanglement-Assisted Quantum Locally Recoverable Codes: Characterizations, Bounds, and Constructions

Yang Li, San Ling, Zhenliang Lu +2 more

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:2607.27091•Jul 29, 2026•0 citations

Bounds for General $(r,δ)$-Quantum Locally Recoverable Codes

Shuowei Ma, Fei Shi

We study general $(r,δ)$-quantum locally recoverable codes (qLRCs) without assuming purity or a stabilizer structure, and with arbitrarily overlapping recovery blocks. Using a selection lemma for multiple-erasure recovery, we establish a Singleton-like upper bound for general quantum subspaces. When $δ=2$, this bound reduces to the general single-erasure bound of Golowich and Guruswami. An integer optimization yields an explicit remainder correction that tightens the bound by up to $δ-2$. For pure codes, local decoupling and entropy arguments extend the parameter bound of Galindo et al., originally obtained for pure stabilizer codes constructed from Euclidean or Hermitian dual-containing classical codes, to arbitrary pure quantum subspaces. We also remove the partition assumption from several existing pure-code bounds for disjoint recovery blocks and obtain a further remainder correction by optimizing the budget. Finally, we analyze three families attaining these bounds: pure codes with overlapping recovery blocks and $d=δ$, which exhibit both a strict improvement of the pure-code bound and simultaneous equality in the pure-code and general bounds; impure codes with disjoint recovery blocks attaining the general bound; and an existing family of pure codes with overlapping recovery blocks and $d>δ$, which simultaneously attains all these bounds.

quant-pharXiv:2610.40273•Oct 30, 2026•0 citations
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.
This paper studies entanglement-assisted quantum locally recoverable codes (EA-qLRCs) built via a CSS-like stabilizer construction from pairs of classical locally recoverable codes (cLRCs), without requiring dual-containment. We define such codes through local recovery channels, give a sufficient stabilizer criterion for the construction, and derive Singleton-, Griesmer-, Plotkin-, and sphere-packing-like converse bounds on the parameters of the resulting pure CSS-like EA-qLRCs, along with a Cadambe--Mazumdar-like bound that, as in the classical case, lacks a closed form, plus a comparison of their relative tightness across finite-length and asymptotic regimes. We give necessary and sufficient conditions for a pure CSS-like EA-qLRC to attain the Singleton-like bound with equality; for the single-code case C1=C2=C\mathcal{C}_1=\mathcal{C}_2=\mathcal{C}C1​=C2​=C, this reduces to a simple condition on the hull dimension s=dim⁡(C∩C⊥)s=\dim(\mathcal{C}\cap\mathcal{C}^\perp)s=dim(C∩C⊥), which also fixes the entanglement count via c=n−k−sc=n-k-sc=n−k−s. We present CSS-like EA-qLRC constructions from classical LRC families---Tamo--Barg and cyclic codes---and characterize when these attain the Singleton-like bound, showing the cyclic families yield optimal codes while the Tamo--Barg construction, though valid, attains the bound only in the degenerate regime k≤rk \le rk≤r, where locality is vacuous. We complement these constructions with two Gilbert--Varshamov-like achievability bounds, via a classical parity-check augmentation and a sharper concatenated-code construction, and show both hold unconditionally for field size q>3q>3q>3 via a monomial-equivalence argument. Finally, we unify all bounds---converse and achievability alike---under a common maximally entangled regime, giving a single comparison of the achievable and forbidden rate--distance--locality region for CSS-like EA-qLRCs.
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) permit reconstruction of a lost coordinate from a small subset of the remaining coordinates, a property that is a fundamental coding technique for distributed storage and, more recently, for fault-tolerant quantum memories and distributed storage systems. Existing constructions of quantum locally recoverable codes (qLRCs) and their entanglement-assisted generalizations (EA-qLRCs) impose a single locality parameter rrr governing recovery from both bit-flip (XXX-type) and phase-flip (ZZZ-type) errors, even though physical noise is rarely symmetric between the two. We introduce entanglement-assisted quantum locally recoverable codes (EA-qLRC) with \emph{asymmetric locality} {rX,rZ}\{r_X,r_Z\}{rX​,rZ​}, in which XXX-type and ZZZ-type errors are corrected from recovery sets of size at most rX+1r_X+1rX​+1 and rZ+1r_Z+1rZ​+1, respectively. We extend a stabilizer sufficient condition for this property and, in particular, the CSS framework, where asymmetric locality corresponds to separate locality conditions on the two classical codes C1\mathcal C_1C1​ and C2\mathcal C_2C2​ underlying the construction. From this characterization we derive Singleton-, Griesmer-, Plotkin-, sphere-packing-, and Cadambe\textendash Mazumdar-like bounds for CSS-type asymmetric EA-qLRCs (AEA-qLRCs) built from Proposition~\ref{CSS-EA-qLRC-Asym}, and we characterize exactly when a pure CSS-type AEA-qLRC built from a pair of classical LRCs attains the Singleton-like bound with equality.
Bounds for Pure Disjoint (r,δ)(r,δ)(r,δ)-Quantum Locally Recoverable Codes
We study pure disjoint (r,δ)(r,δ)(r,δ)-quantum locally recoverable codes (qLRCs) without assuming a stabilizer structure. We formulate local Knill--Laflamme conditions for recovery from up to δ−1δ-1δ−1 erasures within a recovery block, and introduce blockwise Shor--Laflamme and unitary weight enumerators that capture how error weight is distributed across recovery sets. We establish several properties of these enumerators and use them to derive a Singleton-like bound that strengthens the known bound for disjoint (r,δ)(r,δ)(r,δ)-qLRCs under a purity assumption, as well as a linear-programming upper bound on the code dimension. These results provide a non-stabilizer, weight-enumerator-based approach to the study of pure disjoint (r,δ)(r,δ)(r,δ)-qLRCs.
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 rrr 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 ℓ\ellℓ-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.
Bounds for General (r,δ)(r,δ)(r,δ)-Quantum Locally Recoverable Codes
We study general (r,δ)(r,δ)(r,δ)-quantum locally recoverable codes (qLRCs) without assuming purity or a stabilizer structure, and with arbitrarily overlapping recovery blocks. Using a selection lemma for multiple-erasure recovery, we establish a Singleton-like upper bound for general quantum subspaces. When δ=2δ=2δ=2, this bound reduces to the general single-erasure bound of Golowich and Guruswami. An integer optimization yields an explicit remainder correction that tightens the bound by up to δ−2δ-2δ−2. For pure codes, local decoupling and entropy arguments extend the parameter bound of Galindo et al., originally obtained for pure stabilizer codes constructed from Euclidean or Hermitian dual-containing classical codes, to arbitrary pure quantum subspaces. We also remove the partition assumption from several existing pure-code bounds for disjoint recovery blocks and obtain a further remainder correction by optimizing the budget. Finally, we analyze three families attaining these bounds: pure codes with overlapping recovery blocks and d=δd=δd=δ, which exhibit both a strict improvement of the pure-code bound and simultaneous equality in the pure-code and general bounds; impure codes with disjoint recovery blocks attaining the general bound; and an existing family of pure codes with overlapping recovery blocks and d>δd>δd>δ, which simultaneously attains all these bounds.