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Optimal Quantum $(r,\delta)$-Locally Repairable Codes via Classical Ones

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Pith's one-line read Every optimal locally repairable code decomposes into MDS repair blocks, a structure that automatically yields optimal quantum locally repairable codes.

desk verdict Unified decomposition theorem is a genuine advance, but the proof has fixable gaps—worth a serious referee. read the letter →

arxiv 2507.18175 v1 pith:A2MBV3RV submitted 2025-07-24 quant-ph

Kun Zhou , Meng Cao This is my paper · ORCID
classification quant-ph MSC 94B0511T7181P70
keywords locallyrepairablecodes$(r\delta)$-localityoptimaldecompositiontheoremMDSlocalprotectionquantumCSSconstructionHermitian
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Locally repairable codes (LRCs) let a distributed storage system recover a lost symbol by reading only a small local group of other symbols, and optimal LRCs hit the Singleton-like bound on the tradeoff between redundancy and repair locality. This paper proves a unified decomposition theorem: every optimal $(r,\delta)$-LRC is a disjoint union of MDS local protection codes of distance $\delta$ plus a leftover set whose size is exactly the code's minimum distance $d$. Because the decomposition holds in full generality, the paper can show that every local protection code is MDS and that $d\geq \delta$ always holds. That removes a technical condition that previously stood between classical optimal $(r,\delta)$-LRCs and optimal quantum $(r,\delta)$-LRCs, so CSS and Hermitian constructions now convert classical optimal codes directly into optimal quantum ones. The paper also characterizes which optimally decomposed classical codes induce optimal quantum codes and gives three infinite families of such quantum codes.

What carries the argument

The load-bearing mechanism is the subset-selection algorithm (Algorithm 1) operating on the canonical indexed set $\mathcal{C}=\{(i,c_i)\}$. Starting from an empty set, it repeatedly picks a coordinate, chooses a local protection code for that coordinate, and adds the local block (or a sub-block chosen so that the rank increments by exactly one per added vector) until the accumulated rank reaches $k-1$. The key identity is the rank-size inequality for each block: each added local protection set contributes rank at most $|\text{added set}|-(\delta-1)$ but at least $1$, while the terminal leftover set has size at most $r-1$; together with the Singleton-type bound this forces the total size $|S|=k-1+(\lceil k/r\rceil-1)(\delta-1)$ and equality throughout, which is what makes each block MDS. In the quantum half of the paper, the analogous load-bearing object is the block parity-check matrix (4.4), whose off-diagonal blocks determine whether the constituent codes generated by (4.7) are Hermitian or Euclidean self-orthogonal, and hence whether the classical code is dual-containing and induces an optimal quantum LRC.

What would settle it

Exhibit a single optimal $(r,\delta)$-LRC with parameters $[n,k,d]_q$ whose set of generator columns cannot be partitioned as in Case (I) or Case (II) of Theorem 3.4—for instance, an optimal code with a local protection code of length $n_i\le r+\delta-1$ whose punctured distance is $\delta+1$ rather than $\delta$—or an optimal $(r,\delta)$-LRC with $d<\delta$. A brute-force search over small $q$, $r$, $\delta$, $n$, $k$ (e.g. all optimal $(2,2)$-LRCs over $\mathbb{F}_q$ for $q\le 9$) checking whether every repair set's punctured code is MDS would settle the decomposition theorem in those cases.

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Extended reading notes

Core claim

The central discovery is a structural dichotomy for any optimal $(r,\delta)$-LRC with parameters $[n,k,d]_q$. Viewing the code as the indexed set $\mathcal{C}=\{(i,c_i)\}$ of generator-matrix columns, Theorem 3.4 asserts that $\mathcal{C}$ is either $\mathcal{C}=\mathcal{C}_1\cup\cdots\cup\mathcal{C}_{t-1}\cup\{(j_1,c_{j_1}),\ldots,(j_{s_t},c_{j_{s_t}})\}\cup U$ with $t=\lceil k/r\rceil$, or $\mathcal{C}=\mathcal{C}_1\cup\cdots\cup\mathcal{C}_t\cup U$ with $t=\lceil k/r\rceil-1$, where each $\mathcal{C}|_{\mathcal{C}_i}$ is a local protection code that is an MDS code with parameters $[n_i\le r+\delta-1,\, n_i-\delta+1,\, \delta]_q$, the terminal vectors lie inside some local protection code and number at most $r-1$, and the residual set $U$ is disjoint with $|U|=d(\mathcal{C})$. Along the chain the rank-size equations $\operatorname{rank}(\cup_{j=1}^{i}\mathcal{C}_j)-\operatorname{rank}(\cup_{j=1}^{i-1}\mathcal{C}_j)=|\cup_{j=1}^{i}\mathcal{C}_j|-|\cup_{j=1}^{i-1}\mathcal{C}_j|-(\delta-1)$ hold. From this the paper derives that every local protection code of an optimal $(r,\delta)$-LRC is MDS of distance $\delta$ (Theorem 3.7), and that $n-k\ge \lceil k/r\rceil(\delta-1)$, equivalently $d\ge\delta$ (Theorem 4.1(1)). It then proves that an optimal classical $(r,\delta)$-LRC with a minimal decomposition induces an optimal quantum $(r,\delta)$-LRC exactly when its parity-check matrix has the block form (4.4) and all codes generated by the matrices in (4.7) are Hermitian or Euclidean self-orthogonal (Theorem 4.6).

Load-bearing premise

The load-bearing premise is that the algorithm that builds the decomposition never gets stuck—at each step one can pick a repair group that adds new information, and when it stops at most $r-1$ columns remain outside the chosen repair groups—because if a choice forces a larger leftover set, the exact count $|S|=k-1+(\lceil k/r\rceil-1)(\delta-1)$ that drives the theorem fails.

Editorial extensions

If this is right

  • Every local protection code of an optimal $(r,\delta)$-LRC is an MDS code with parameters $[n_i\le r+\delta-1,\, n_i-\delta+1,\, \delta]_q$, so local repair sets are as efficient as possible.
  • Every optimal $(r,\delta)$-LRC satisfies $n-k\ge \lceil k/r\rceil(\delta-1)$, i.e. $d\ge\delta$, guaranteeing that the code can always correct at least $\delta-1$ erasures.
  • Any Hermitian dual-containing (resp. Euclidean dual-containing) optimal $(r,\delta)$-LRC induces, through the Hermitian (resp. CSS) construction, an optimal quantum $(r,\delta)$-LRC with no separate inequality check needed.
  • An optimal classical $(r,\delta)$-LRC with a minimal decomposition induces an optimal quantum LRC if and only if the associated block codes are all Hermitian or all Euclidean self-orthogonal, giving a complete and checkable criterion.
  • Three infinite families of optimal quantum $(r,\delta)$-LRCs exist with flexible parameters, including one family whose length grows super-linearly in the field size.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • An implication the authors leave implicit: the decomposition theorem suggests that repair groups of an optimal $(r,\delta)$-LRC form near-disjoint MDS islands, so a storage system could schedule repairs within each island independently without global coordination; this is directly testable on existing optimal constructions.
  • The $d\ge\delta$ bound, combined with the quantum Singleton-type bound, implies these constructions can never produce quantum LRCs with $d<\delta$; if applications need shorter distances, one would have to step outside the optimal classical class or weaken locality.
  • A testable extension is to apply the parity-check criterion of Theorem 4.6 to known families of optimal LRCs (pyramid codes, Tamo-Barg codes, propagated constructions) and enumerate which of them become optimal quantum LRCs, which would produce many new explicit parameter sets beyond the three families given.
  • The minimal-decomposition concept could be made algorithmic: finding the minimal decomposition of a given optimal code is a combinatorial optimization problem of choosing local blocks maximizing rank gain per coordinate, and automating it would let one certify optimal quantum LRCs by computer search.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

5 major / 5 minor

Summary. The paper studies optimal (r,δ)-locally repairable codes and their quantum counterparts. Its main structural claim (Theorem 3.4) is that every optimal (r,δ)-LRC, viewed as an indexed set of generator-matrix columns, admits one of two decompositions into MDS local protection codes, a terminal set, and a residual set whose size equals the minimum distance. From this decomposition the paper derives rigidity results (every local protection code is MDS, Theorem 3.7; d ≥ δ, Theorem 4.1(1)), a parity-check criterion for optimal quantum (r,δ)-LRCs induced by classical optimal codes admitting a minimal decomposition (Theorem 4.6), and three infinite families of optimal quantum (r,δ)-LRCs with explicit parameters (Theorems 5.1, 5.4, 5.6).

Significance. If the decomposition theorem is established, it is a substantial structural contribution: it extends earlier partition-type results of Song et al. and Prakash et al. to the general case, including r | (k−1), and it gives a clean route to quantum (r,δ)-LRC constructions while removing auxiliary conditions present in Galindo et al. The paper is largely self-contained and the three construction families are concrete, with worked examples in Section 5. The main weakness is that the proof of the central decomposition rests on an incompletely specified selection algorithm, and several secondary structural claims are asserted rather than proved. With a repair of Lemma 3.3, the central results are plausible and potentially publishable.

major comments (5)
  1. [§3.1, Lemma 3.3 (Algorithm 1)] Step P2 of Algorithm 1 chooses (i_j,c_{i_j}) from C \ S_{j-1}, i.e., an unselected indexed element, not a vector outside span(S_{j-1}). Step P4 then adds a full local protection code S_j whenever rank(S_{j-1} ∪ S_j) < k, even if the rank increase is zero, which can happen when the newly picked column lies in the current span and the chosen local code for it is contained in that span. The proof asserts termination and that each added block satisfies condition (2) of Lemma 3.1 (rank increment at least 1), but this is not guaranteed by the stated algorithm; a run could add zero-rank blocks, invalidating the accounting |S| = k−1+(⌈k/r⌉−1)(δ−1) and leaving the terminal-size bound s_{i0+1} ≤ r−1 unproved. This is load-bearing: Lemma 3.3 is used to prove Theorem 3.4, Theorem 3.7, and Theorem 4.1(1). The likely fix is local: while rank < k, always pick a column outside the current span (such a column exists because the generator matrix has rank k), and then prove that every added block increases the rank by at least one and that the terminal set satisfies the stated bound.
  2. [§3.2, Theorem 3.7] The proof's statement that 'we can require that C1 in Case (I) is exactly S1 due to the proof of Theorem 3.4' is not an argument. Theorem 3.4, as stated, produces one decomposition; to show that an arbitrary local protection code C|S1 is MDS, one must rerun the corrected selection procedure with S1 as the first chosen local protection code and verify that the rank-growth and terminal-set conditions still hold. As written, Theorem 3.7 is an unproved invariance claim rather than a consequence of Theorem 3.4.
  3. [§3.2, Remark 3.10] The observation that any two distinct local protection codes satisfy S1 = S2 or S1 ∩ S2 = ∅ is false. The second code in Example 3.6 is verified there to be an optimal (2,2)-LRC, and its two displayed local protection codes C|{1,2,3} and C|{2,4,5} are distinct yet intersect in coordinate 2. This remark should be corrected or removed; it is not needed for the decomposition theorem, and as stated it gives a false consistency claim with [28, Theorem 9].
  4. [§4.1, Theorem 4.1(1) proof] After defining C_t = C|T for a local protection code T containing the terminal set, the proof asserts 'By Case (I) of Theorem 3.4, we have s_i ≥ δ' for all i ∈ [t]. For i ≤ t−1 this follows from condition (2), but for i = t it does not: s_t is the number of new coordinates contributed by T outside the earlier blocks, and Theorem 3.4's Case (I) only says that the terminal set is contained in some local protection code T. The argument needs the additional condition rank(C_1 ∪ ... ∪ C_{t−1} ∪ T) = k, which appears in the construction inside Lemma 3.3 but is omitted from the statement of Theorem 3.4, together with a proof that this condition and the Singleton bound on T imply s_t ≥ δ. Without this, Theorem 4.1(1), and hence d ≥ δ, is not fully established.
  5. [§4.2, Theorem 4.3] The proof of Theorem 4.3 is incomplete in its rank computation. The sentence 'the number of repeated indices between cH_i and H_{i+1} is less than n_i−(δ−1)' uses an undefined quantity and is the only justification given for the claim that the matrix in Eq. (4.5) has rank t(δ−1). The subsequent extension by appending B_1,...,B_t also needs a proof that such matrices exist and that together with the top blocks they have rank t(δ−1)+l. Because Theorem 4.6, Corollary 4.7, and the constructions in Section 5 depend on this parity-check characterization, the proof should be completed.
minor comments (5)
  1. [§3.2, Corollary 3.11] The displayed identity ends with '−δ−1'; it should presumably be '−(δ−1)' to match Lemma 3.1. The proof sketch is also too terse to verify the claimed conclusion.
  2. [§4.2, Theorem 4.3 proof] The reference 'Corollary 3.7' should be 'Theorem 3.7'.
  3. [§5.3, Example 5.8] The sentence 'the code C in Theorem 5.4 has a parity-check matrix' should refer to Theorem 5.6.
  4. [§2.3, Definition 2.8] The expression 'dim(C) = n+k/2' should be 'dim(C) = (n+k)/2' to be consistent with the Hermitian construction parameters.
  5. [General notation] Using the same symbol C for both the linear code and the indexed set of its generator-matrix columns can be confusing in statements such as Theorem 4.3 and Corollary 4.4; a distinct symbol for the indexed set would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the decomposition, rigidity, and quantum-LRC results are derived within the paper from definitions, Lemma 2.4, and the Singleton-type bound, with no fitted inputs or load-bearing self-citations.

full rationale

The paper's central claim, Theorem 3.4, is not circular. Its proof is carried out inside the paper: Lemma 3.1 and Lemma 3.2 start from an assumed subset S with rank k-1, local protection sets, and rank-growth inequalities; using Lemma 2.4 (an independent, standard rank characterization of minimum distance) and the Singleton-type bound for (r,delta)-LRCs, they force the size of S and the MDS property of the first local block. Lemma 3.3 then tries to produce such an S by Algorithm 1. There is a genuine rigor gap here: Step P2 chooses from the indexed-set complement rather than the linear span, so a P4 step need not increase rank, and the proof does not fully justify that every local protection code can be placed first. However, this is a proof-gap/correctness issue, not a circularity: the desired decomposition is not assumed in the hypotheses, and no quantity is fitted or renamed. The later rigidity results (Theorem 3.7, Corollary 3.9, Proposition 3.8) are corollaries of Theorem 3.4, not inputs to it. Theorem 4.1 derives n-k >= ceil(k/r)(delta-1) from Theorem 3.4 plus the MDS parity-check matrices of the local blocks; Lemma 2.9 is a separate sufficient condition. Theorem 4.3 and Theorem 4.6 are equivalences between a parity-check form and the relevant optimality/self-orthogonality conditions; they are verified directly, not by assuming the conclusion. The constructions in Theorems 5.1, 5.4, and 5.6 are explicit and verified by direct Hermitian-inner-product computations. There is no load-bearing self-citation in the critical path; references [7], [19], and [23] are used as standard external lemmas. Thus the circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper's derivations rest on standard coding-theoretic results: the generalized Singleton bound for (r,δ)-LRCs [23], the parity-check characterization [19, Lemma 2], and the CSS/Hermitian constructions [3, 31, 1]. No ad hoc assumptions or fitted parameters are introduced; the construction parameters (q,u,v,t,s) are free design choices with stated constraints, not fitted values.

assumptions (4)
  • domain assumption Generalized Singleton-type bound for (r,δ)-LRCs: d ≤ n-k+1-(⌈k/r⌉-1)(δ-1)
    Used as the definition of optimality for (r,δ)-LRCs; cited from [23] and invoked throughout, especially in Lemmas 3.1-3.3 and Theorem 4.1.
  • domain assumption Parity-check characterization of (r,δ)-locality (Lemma 2.5)
    Imported from [19, Lemma 2]; used to connect locality to the existence of sparse parity-check matrices, foundational for Theorem 4.1 and the constructions in Section 5.
  • standard math CSS construction and Hermitian construction for quantum codes
    Propositions 2.6 and 2.7 are accepted background results from [3, 31, 1]; used to turn classical dual-containing codes into quantum codes.
  • standard math Standard linear algebra over finite fields
    Rank, dimension, generator and parity-check matrix duality, punctured codes, and the distance characterization Lemma 2.4 are standard and used throughout.

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Pith. "Pith review of Optimal Quantum $(r,\delta)$-Locally Repairable Codes via Classical Ones." pith.science (2026). https://pith.science/paper/A2MBV3RV

@misc{pith2026250718175,
  author       = {Pith},
  title        = {Pith review of: Optimal Quantum $(r,\delta)$-Locally Repairable Codes via Classical Ones},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/A2MBV3RV}},
  note         = {Machine review of arXiv:2507.18175}
}
abstract

Locally repairable codes (LRCs) play a crucial role in mitigating data loss in large-scale distributed and cloud storage systems. This paper establishes a unified decomposition theorem for general optimal $(r,\delta)$-LRCs. Based on this, we obtain that the local protection codes of general optimal $(r,\delta)$-LRCs are MDS codes with the same minimum Hamming distance $\delta$. We prove that for general optimal $(r,\delta)$-LRCs, their minimum Hamming distance $d$ always satisfies $d\geq \delta$. We fully characterize the optimal quantum $(r,\delta)$-LRCs induced by classical optimal $(r,\delta)$-LRCs that admit a minimal decomposition. We construct three infinite families of optimal quantum $(r,\delta)$-LRCs with flexible parameters.

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Forward citations

Cited by 2 Pith papers

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