REVIEW 3 major objections 3 minor 4 cited by
On optimal quantum LRCs from the Hermitian construction and $t$-designs
T0 review · 3 major / 3 minor · reviewed 2026-08-05 · deepseek-v4-flash
Pith's one-line read The paper establishes four bounds for quantum locally recoverable codes and constructs three explicit families of optimal codes meeting them, using the Hermitian construction and classical codes that support 2- and 3-designs.
desk verdict Abstract-only submission with plausible, meaningful claims: four qLRC bounds, NMDS codes supporting t-designs, and optimal qLRC families—but the central optimality claims rest on proofs and parameter tables we cannot see. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The Hermitian construction is the conversion that turns a classical code with Hermitian dual-containment into a quantum code; the engine is a supply of NMDS codes with flexible dimensions that support $t$-designs for $t\in\{2,3\}$. These classical codes simultaneously give the locality structure and the duality needed for the quantum conversion, while the four new bounds provide the target against which optimality is measured.
What would settle it
Take the smallest nontrivial code from any of the three families and compute its exact distance, locality, and dual-containment property; if a code violates its claimed bound or the Hermitian dual-containment fails, the central claim collapses. Alternatively, exhibit any qLRC whose parameters contradict one of the four bounds.
Extended reading notes
Core claim
The central claim is that the parameter region for qLRCs is sharply constrained by four bounds, and that these bounds are simultaneously achievable. The authors build new infinite families of NMDS codes whose dimensions can be freely chosen and which support designs of strength 2 and 3. These classical codes are used to obtain Hermitian dual-containing classical LRCs, and the Hermitian construction turns them into qLRCs. Three explicit families of qLRCs are shown to be optimal, meaning they meet the relevant bound; this resolves an open problem in the recent literature. As a by-product, the underlying cLRCs are themselves optimal with respect to four different cLRC bounds.
Load-bearing premise
The whole construction is optimal only if the four proposed bounds are valid and tight and if the Hermitian construction really delivers codes with the claimed distance and locality parameters.
Editorial extensions
If this is right
- Three explicit infinite families of qLRCs achieve optimal parameters under the new bounds, so the bounds are not just theoretical.
- The new qLRC families allow more flexible dimensions than earlier CSS-based constructions, widening the choice of parameters for quantum storage codes.
- The classical LRCs produced in the process are optimal with respect to four distinct bounds, so the construction has independent classical value.
- The open problem about the existence of optimal qLRCs from the Hermitian construction is resolved affirmatively.
Reading between the lines
- If the four bounds hold in their full generality, existing qLRC constructions in the literature should be re-checked against them; some may be suboptimal in regimes not covered by the paper's examples.
- The fact that the classical ingredients support $2$- and $3$-designs suggests a combinatorial structure that could be exploited for erasure-recovery scheduling or for constructing quantum codes with transversal gates, but the paper does not develop this.
- A natural extension would be to apply the same NMDS-plus-$t$-design recipe to other duality-preserving constructions (for example, CSS with varied field sizes) to see whether optimal qLRC families exist beyond the three reported here.
- The asymptotic comparison of the four bounds likely indicates which bound is dominant in different blocklength regimes, which could guide code designers before the full parameter tables are computed.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper claims to introduce and prove four bounds for quantum locally recoverable codes (qLRCs), to compare their asymptotic behavior, to construct new infinite families of NMDS codes supporting t-designs for t=2,3, to apply these codes to obtain Hermitian dual-containing classical LRCs, and to derive three explicit families of optimal qLRCs. It also states that this solves an open problem posed by Luo et al. The abstract does not include the statements of the bounds, the parameters of the constructed codes, the verification of dual containment, or the equality conditions under which optimality is certified.
Significance. If the claims are correct, the paper would be a substantial contribution to quantum coding theory: new bounds with asymptotic comparison, a systematic Hermitian construction producing dual-containing classical LRCs, and optimal qLRC families with more flexible parameters than CSS-based constructions. The construction of NMDS codes supporting t-designs is independently interesting. However, the significance can only be provisionally assessed from the abstract; the central optimality claims are not independently checkable without the full derivations and parameter tables.
major comments (3)
- [Abstract] The four qLRC bounds are stated only by name ('we present four bounds') with no equations, no parameter regimes, and no hypotheses on alphabet size, locality, or block length. Correctness and tightness of these bounds are load-bearing for the optimality claims. The abstract does not even specify whether the bounds are the same as or distinct from the prior cLRC bounds referenced later. The full manuscript must include precise theorem statements and proofs; without them the central claim cannot be evaluated.
- [Abstract] The claim of 'three explicit families of optimal qLRCs' is not accompanied by any parameter values (length, dimension, distance, locality, alphabet size) or by the equality conditions to the bounds. Optimality requires both that the bounds are valid and that the constructions attain them in the stated regimes. As written, this is an assertion rather than a verifiable result. A parameter table and an explicit check of the equality conditions for each family are needed.
- [Abstract] The phrase 'Hermitian dual-containing classical LRCs' is used as a step in the construction, but the abstract does not show how the NMDS/t-design codes guarantee dual containment, nor how locality and distance of the resulting cLRCs are computed. Since the qLRC construction depends on these properties, an omitted proof or even an omitted definition of the Hermitian construction would leave a load-bearing gap. The full text must provide the explicit construction and the verification of the cLRC parameters.
minor comments (3)
- [Abstract] The abstract does not fix notation for qLRC parameters (e.g., [[n,k,d;r]]_q or similar). Adding standard notation would improve clarity and allow readers to parse the optimality claims.
- [Abstract] The notion of t-designs is used without definition or reference. Since the NMDS-code construction is a stated contribution, a brief definition or a pointer to the standard definition would help.
- [Abstract] The reference to Luo et al. is cited with full bibliographic data in the abstract, which is unusual; if the paper is intended for journal submission, the open problem should be stated explicitly rather than only cited.
Circularity Check
No circularity detectable from the abstract; the claims rest on unshown derivations and constructions, but there is no evidence of self-referential reasoning or fitted-input prediction.
full rationale
This is an abstract-only review. The abstract reports four bounds for qLRCs, constructions of NMDS codes supporting t-designs, Hermitian dual-containing classical LRCs, and three families of optimal qLRCs. No equations, parameter tables, or proof sketches are provided. To claim circularity, the instructions require quoting the paper and exhibiting a specific reduction (e.g., a parameter fitted to the same data then renamed a prediction, or a definition that embeds the result). The abstract contains no such equations or constructions. The reference to Luo et al. is an acknowledgement of an open problem, not a load-bearing self-citation. The optimality claims are unverified from the abstract, but unverified is not the same as circular. Therefore, the appropriate finding is no significant circularity, score 0.
Assumptions & free parameters
Cite this review
Pith. "Pith review of On optimal quantum LRCs from the Hermitian construction and $t$-designs." pith.science (2026). https://pith.science/paper/A4XZDUP6
@misc{pith2026250813553,
author = {Pith},
title = {Pith review of: On optimal quantum LRCs from the Hermitian construction and $t$-designs},
year = {2026},
howpublished = {\url{https://pith.science/paper/A4XZDUP6}},
note = {Machine review of arXiv:2508.13553}
}
abstract
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.
Forward citations
Cited by 4 Pith papers
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Quantum Hierarchical Locally Recoverable Codes
Random and explicit (r,δ) quantum LRCs and h-level hierarchical QLRCs are constructed via CSS dual-containing codes, with distance bounds, a Singleton-like bound, and an efficient decoder for the Tamo–Barg family.
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CSS Quantum LRCs with Intersecting Recovery Sets: Constructions and Bounds
CSS quantum locally recoverable codes with intersecting recovery sets are characterized by classical codes with common recovery sets, and explicit binary families with high rates are constructed.
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Entanglement-Assisted Quantum Locally Recoverable Codes: Bounds, Optimal Constructions, and Achievability
Entanglement-assisted quantum locally recoverable codes can be constructed from arbitrary classical LRC pairs, and this paper proves bounds, optimality conditions, and explicit constructions for them.
-
Bounds for Pure Disjoint $(r,\delta)$-Quantum Locally Recoverable Codes
New Singleton-like and linear-programming dimension bounds are proven for disjoint quantum locally recoverable codes under a blockwise purity condition.
Reviewed August 5, 2026 · model on record in the stance chip above.
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