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Extension of Optimal Locally Repairable codes

T0 review · 4 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read Extending one coordinate of an optimal LRC preserves optimality, pushing rational-field lengths to q+2 and improving elliptic-field LRCs.

desk verdict A solid incremental paper: the rational-function-field extension to length q+2 and the (r,3)-LRCs are real, but the elliptic (r,3)-LRC claim rests on an unproved and possibly false invertibility condition. read the letter →

arxiv 2501.18989 v1 pith:F3ODCT2O submitted 2025-01-31 cs.IT math.IT

Yunlong Zhu , Chang-An Zhao This is my paper · ORCID
classification cs.ITmath.IT MSC 94B0594B2711T7114H52
keywords locallyrepairablecodesoptimalLRCfunctionfieldsellipticcurvesrational(r3)-LRCSingleton-typeboundextended
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

This paper claims that optimal locally repairable codes (LRCs) can be extended by a single coordinate attached to one repair block without losing either locality or optimality. For codes built from rational function fields, the extension pushes the maximal achievable length from q+1 to q+2 over $\mathbb{F}_q$. For optimal LRCs from elliptic function fields, the same one-coordinate extension preserves optimality and yields lengths up to $q+2\sqrt{q}-2r-2$ with locality $r$. The paper also constructs $(r,3)$-LRCs by extending every block, with optimality when the number of blocks equals the dimension parameter $t$.

What carries the argument

The machinery is the function-space evaluation code together with a one-coordinate extension that records the top coefficient of the evaluated function. For rational fields, the function space $V=\{\sum_{i=0}^{r-1}\sum_{j=1}^t a_{i,j}f_jz^i\}$ has dimension $rt$, and the appended symbol $a_{r-1,t}$ is constant on the selected block, so any $r$ values of that block recover it by the same Lagrange interpolation that recovers the other symbols; the minimum-distance proof splits on whether $a_{r-1,t}$ is zero. For elliptic fields, the space $V_{E,1}$ replaces part of the basis by powers of $\hat z$ so that the appended coordinate $a_{0,t}$ is again recoverable from the block values and the distance bound follows from the divisor $G=(t-1)(P_1+\cdots+P_{r+1})$.

What would settle it

Take $q=64$ and an elliptic curve over $\mathbb{F}_{64}$ with automorphism group of order $24$, choose $r=7$ and a block of points as in Section 2.5, and compute the $6\times 6$ minors of the matrix $M'_u$; if any of these $(r-1)\times(r-1)$ minors vanishes for a block that satisfies the $r\times r$ condition of [22, Lemma 4], then the hypothesis of Theorem 13 fails and the constructed $(r,3)$-LRC is not guaranteed by the paper's proof.

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Extended reading notes

Core claim

The central discovery is that if $C$ is an optimal LRC whose repair blocks are each $[r+1,r]$ MDS codes, then appending one extra coordinate that records a carefully chosen coefficient, the top-degree term $a_{r-1,t}$ of the defining function, to one block yields a code that is still an optimal LRC. In the rational-function-field setting this gives optimal LRCs of length $s(r+1)+1$, dimension $rt$ and locality $r$ for the three listed conditions on $r$ and $2\le s\le (q+1-2r)/(r+1)$; when $r+1$ divides $q+1$, the length reaches $q+2$. The same mechanism, using the function space $V_{E,1}$ built from a rational function $\hat z$ with $(\hat z)=Q_1-Q_\infty$, extends optimal elliptic LRCs to length $s(r+1)+1$ and preserves optimality for $s\le \lfloor(N-2r-4)/(r+1)\rfloor$ on a curve with $N$ rational points. Extending all $s$ blocks instead produces $(r,3)$-LRCs of length $s(r+2)$, with optimality when $t=s$.

Load-bearing premise

For the elliptic $(r,3)$-construction, every $(r-1)\times(r-1)$ submatrix of the block evaluation matrix $M'_u$ must be invertible, a stronger condition than the $r\times r$ condition the paper inherits from the prior elliptic construction, and the paper neither proves it nor cites a construction that guarantees it.

Editorial extensions

If this is right

  • Over $\mathbb{F}_q$, optimal LRCs with locality $r$ exist with length $q+2$ whenever $r+1$ divides $q+1$, exceeding the previous $q+1$ bound for rational-function-field constructions.
  • On an elliptic curve with $N$ rational points, the single-coordinate extension gives optimal LRCs of length $s(r+1)+1$ for $s\le \lfloor(N-2r-4)/(r+1)\rfloor$, reaching $q+2\sqrt{q}-2r-2$.
  • Extending every block yields optimal $(r,3)$-LRCs of length $s(r+2)$ and dimension $rt$ in the rational case when $t=s$, meeting the corresponding Singleton-type bound.
  • The Roth-Lempel type extension gives LRCs with distance at least $n-rt-t+1$, one below optimal, and reaches optimal under extra element-sum conditions.
  • The extension idea applies to any LRC whose repair blocks are MDS codes, not only those arising from function fields.

Reading between the lines

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

  • If the extension operation is applied to a different block after the first, the appended coordinate would generally break the locality argument; a testable question is whether sequential extension of $m$ distinct blocks preserves optimality for $m>1$, and for which $m$.
  • The unproved $(r-1)\times(r-1)$ minor condition in the elliptic $(r,3)$ construction could be checked computationally on curves with known automorphism groups; if it fails generically, the construction may require selecting special block points.
  • The extension view recasts the added coordinate as a parity node attached to a repair set; in distributed-storage terms this suggests repair schemes where one extra node holds a linear function of a repair group's data.
  • One could test whether the rational-field length bound $q+2$ is tight for this method: the same argument does not immediately extend to $q+3$ because two appended coordinates would need to be simultaneously recoverable.
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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

4 major / 5 minor

Summary. The paper studies optimal locally repairable codes (LRCs) constructed from function fields. It proposes a general extension mechanism: given an optimal LRC in which every repair block is an [r+1,r] MDS code, adding one coordinate to a block, or one coordinate to every block, preserves optimality or yields an optimal (r,3)-LRC. The main constructions cover rational function fields (extending the codes of Barg–Tamo–Vladut and Jin–Ma–Xing to length q+2), Roth–Lempel type extensions, and elliptic function fields (extending the codes of Li–Ma–Xing to length about q+2√q, plus a conditional (r,3)-elliptic construction). Theorem 5, Theorem 6, Theorem 7, and Theorem 11 contain the core extension arguments; Theorem 13 is conditional on an unproved matrix invertibility condition. Section 4 claims optimality for Roth–Lempel type extensions, but the formal statement in Corollary 9 proves only a weaker distance bound.

Significance. If the main theorems are correct, the extension idea is simple and useful: it increases the known maximal length of optimal LRCs by one coordinate in both the rational and elliptic cases, while preserving locality and optimality. The proofs of the rational-field results (Theorems 5–7) are largely checkable and constructive, and the paper gives explicit parameter examples. However, two load-bearing gaps remain. First, the elliptic (r,3)-LRC construction in Section 5.2 depends on a stronger-than-known invertibility condition that is never proved or instantiated. Second, the Roth–Lempel section claims optimality 'under certain conditions' without stating them, and Corollary 9 only establishes a distance bound one below the Singleton-type bound. These issues prevent the paper from being accepted in its present form, but they are local and may be repairable within the manuscript's scope.

major comments (4)
  1. [Section 5.1, Theorem 11] The proof of locality for the appended coordinate a_{0,t} assumes that z^{t-1}(Q_1) is nonzero, since c_0 = a_{0,t} z^{t-1}(Q_1) must be divided by z^{t-1}(Q_1) to recover a_{0,t}. The function z has a zero somewhere in F, and if Q_1 is chosen to be that zero then the appended symbol is invisible in the first block, so locality fails. The theorem statement does not require z(Q_1) ≠ 0, nor does it prove that Q_1 can be chosen to avoid the zero of z while still having enough splitting-complete places. This condition must be added and its existence established for the length q+2√q−2r−2 claim to hold.
  2. [Section 5.2, Lemma 12 and Theorem 13] Lemma 12 requires that every (r−1)×(r−1) submatrix of M'_u is invertible. This is strictly stronger than the property proved in [22, Lemma 4], which only asserts that every r×r submatrix of M_u = [1, ω_1, …, ω_{r−1}] is invertible. For r=2, Lemma 4 allows three distinct values of ω_1, including a possible zero value, but the new condition additionally requires all three values to be nonzero. The paper neither proves that the automorphism-group construction yields such a point set nor gives an explicit example satisfying the condition. If no such set exists, the claimed optimal (r,3)-elliptic LRC family is empty, so Theorem 13 remains a conditional statement with an unverified hypothesis.
  3. [Section 4, Corollary 9] The claimed Roth–Lempel optimality is not established. Corollary 9 proves only d ≥ n−rt−t+1, which is one less than the Singleton-type bound d ≤ n−rt−t+2 for an [n, rt] LRC with locality r. The text after the proof says optimality holds 'if the elements satisfying some conditions', but those conditions are never stated. In addition, the code C_{RL,1} is defined with two appended coordinates (a_{r−2,t}, a_{r−1,t}), so its length should be s(r+1)+2, not s(r+1)+1 as written in Corollaries 9 and 10. The distance analysis must explicitly account for both appended symbols before any optimality claim can be made.
  4. [Corollary 8] The stated length n=(s+1)(r+1) is inconsistent with the displayed definition of C^m_e. Counting the coordinates in its defining tuple gives s(r+1)+s (or more, depending on the intended appended symbols), not (s+1)(r+1). This inconsistency affects the validity of the claimed d-optimality, since the Singleton-type bound depends on n. The definition and the length must be corrected, and the optimality proof should be re-run with the correct parameters.
minor comments (5)
  1. [Example 1] In the paragraph for r=3, the code is described as having 'locality 4', but the construction has locality r=3; this is a typo.
  2. [Example 2] The sentence 'by Theorem 7' should read 'by Theorem 6', since it refers to the rational function field length-q+2 construction.
  3. [Theorem 13 proof] In the proof of Theorem 13, the text says 'the length and dimension of the code CE are indeed s(r+1)+1 and rt−r+1', but the code CE has length s(r+2); this should be corrected.
  4. [Theorem 5] Condition 3 in Theorem 5 is stated as r+1 = u p^v; the paper should specify that u is an integer coprime to p, since otherwise the condition is ambiguous.
  5. [Section 5] The notation Q_∞, P_∞ in Section 5 is introduced without a clear connection to the places Q_0, P_{0,1}, …, P_{0,r+1} used in Section 2.5; aligning this notation would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular dependence detected: the extension constructions use external optimal LRC families as inputs and prove optimality against the Singleton-type bound by direct pole/zero degree estimates.

full rationale

The paper's derivation chain is not circular in the sense defined here. Its central constructions take the optimal LRC families of [18] and [22] as external starting points, augment the evaluation space with one or more explicitly defined new coordinates, and then prove locality and optimality by direct arguments: locality is shown by Lagrange interpolation using the known MDS property of the blocks, and optimality is shown by bounding deg(fa)∞ and comparing the resulting distance lower bound with the Singleton-type upper bound. For example, in Theorem 5 the bound d ≥ n − rt − t + 2 is obtained from zero-counting on fa, not from the already-known distance of the input code; similarly, Theorem 11 derives its distance bound from the divisor degree of the new function space VE,1. The extra hypothesis in Lemma 12 and Theorem 13, that every (r−1)×(r−1) submatrix of M'u is invertible, is an explicitly stated new assumption rather than a conclusion reused from the theorem being proved; whether it is satisfiable by the elliptic construction is a correctness and instantiation question, not a circularity. The only self-citation, reference [12] by Chen, Mesnager, and Zhao, appears in a related-work list and is not load-bearing for any claim in the paper. The paper even notes in Remark 2 that the Roth-Lempel type (r,3)-extension is not optimal, which is inconsistent with a pattern of forcing the target conclusion by definition. Accordingly, no circular step is exhibited and the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

Central claims rely on standard Hurwitz and Riemann-Roch theory and on existence results imported from [18] and [22]. The only new ad hoc assumption is the M'_u non-singularity condition in Theorem 13. No parameters are fitted to data.

assumptions (5)
  • standard math Hurwitz Genus Theorem and Riemann-Roch theorem are used to bound ramified places and dimensions.
    Invoked in the proofs of Theorems 5, 6 and 11 to count splitting places and to compute dimensions of Riemann-Roch spaces.
  • domain assumption Existence of automorphism subgroups of rational function fields of order r+1 under the stated conditions on r and q.
    Imported from [18] and used in Theorems 5 and 6 to form the extension E/F of degree r+1.
  • domain assumption Lemma 4 from [22]: existence of functions z and ω_i such that every r×r submatrix of M_u is invertible.
    Used in the elliptic constructions of Theorems 11 and 13 to establish locality on each block.
  • ad hoc to paper For Theorem 13, every (r−1)×(r−1) submatrix of M'_u is invertible.
    Assumed without proof or construction; this is the main extra condition needed for (r,3)-locality in the elliptic case.
  • standard math Hasse bound on #E(Fq) is used to translate the elliptic length bound into q+2√q−2r−2.
    Standard bound on the number of rational points of an elliptic curve over a finite field.

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Pith. "Pith review of Extension of Optimal Locally Repairable codes." pith.science (2026). https://pith.science/paper/F3ODCT2O

@misc{pith2026250118989,
  author       = {Pith},
  title        = {Pith review of: Extension of Optimal Locally Repairable codes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/F3ODCT2O}},
  note         = {Machine review of arXiv:2501.18989}
}
abstract

Recent studies have delved into the construction of locally repairable codes (LRCs) with optimal minimum distance from function fields. In this paper, we present several novel constructions by extending the findings of optimally designed locally repairable codes documented in the literature. Let $C$ denote an optimal LRC of locality $r$, implying that every repairable block of $C$ is a $[r+1, r]$ MDS code, and $C$ maximizes its minimum distance. By extending a single coordinate of one of these blocks, we demonstrate that the resulting code remains an optimally designed locally repairable code. This suggests that the maximal length of an optimal LRC from rational function fields can be extended up to $q+2$ over a finite field $\mathbb{F}_q$. In addition, we give a new construction of optimal $(r, 3)$-LRC by extending one coordinate in each block within $C$. Furthermore, we propose a novel family of LRCs with Roth-Lempel type that are optimal under certain conditions. Finally, we explore optimal LRCs derived from elliptic function fields and extend a single coordinate of such codes. This approach leads us to confirm that the new codes are also optimal, thereby allowing their lengths to reach $q + 2\sqrt{q} - 2r - 2$ with locality $r$. We also consider the construction of optimal $(r, 3)$-LRC in elliptic function fields, with exploring one more condition.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

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

    quant-ph 2025-07 conditional novelty 7.0 of 10

    Optimal (r,δ)-locally repairable codes decompose into MDS local codes, forcing d≥δ, and this structure yields new optimal quantum (r,δ)-LRCs.

Reference graph

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25 extracted references · 25 canonical work pages · cited by 1 Pith paper

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