Locally Repairable Codes with Availability via Elliptic Function Fields

Chang-An Zhao; Junjie Huang

arxiv: 2605.06182 · v1 · submitted 2026-05-07 · 💻 cs.IT · math.IT

Locally Repairable Codes with Availability via Elliptic Function Fields

Junjie Huang , Chang-An Zhao This is my paper

Pith reviewed 2026-05-08 04:57 UTC · model grok-4.3

classification 💻 cs.IT math.IT

keywords locally repairable codesavailabilityelliptic function fieldsordinary elliptic curvesrecovering setsfinite fieldsSingleton defect

0 comments

The pith

Ordinary elliptic curves yield several families of optimal q-ary locally repairable codes with length O(q + 2√q) and flexible locality.

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

The paper shows how to build locally repairable codes with one or two recovering sets by drawing on elliptic function fields from ordinary elliptic curves that possess many rational points. Earlier constructions relied mainly on supersingular curves; switching to ordinary curves produces new optimal q-ary examples whose lengths grow like q plus twice the square root of q, together with a general framework that also delivers q-squared-ary codes with two recovering sets and small Singleton defect. A reader would care because these codes directly support efficient local repair and data availability in large distributed storage systems while expanding the algebraic tools available for code design.

Core claim

By employing ordinary elliptic curves with approximately q + 2√q rational points instead of maximal supersingular ones, the authors derive q-ary optimal locally repairable codes with length O(q + 2√q) and flexible locality. They also introduce a general framework using automorphism groups of elliptic function fields to construct locally repairable codes with two recovering sets, realizing it with both supersingular and ordinary curves to achieve a family of q²-ary codes of length O(q² + 2q) with Singleton-defect O(2ℓ / (q² + 2q - 8ℓ)) where ℓ divides q + 2 and 4ℓ < q.

What carries the argument

Automorphism groups of elliptic function fields, harnessed to define the auxiliary functions e_i that produce the recovering sets in the code construction.

If this is right

Optimal q-ary LRCs become available at lengths around q + 2√q with flexible locality parameters.
Codes with two recovering sets achieve lengths O(q² + 2q) over q²-ary alphabets with controlled Singleton defect.
The approach broadens the curves usable for optimal LRC constructions beyond supersingular elliptic curves.
New families arise for both one and two recovering sets via ordinary and supersingular curves.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

These parameters could reduce repair bandwidth in cloud storage while keeping data available even after one repair set is lost.
The same automorphism technique may adapt to other algebraic function fields to produce codes over different alphabets or with different locality profiles.
Explicit e_i constructions supply a reusable template for turning geometric automorphisms into multiple-repair-set codes.

Load-bearing premise

Ordinary elliptic curves over finite fields exist with enough rational points and suitable automorphism groups to define the required auxiliary functions for the recovering sets.

What would settle it

An explicit small-q example where the codes built from an ordinary elliptic curve fail to achieve the claimed minimum distance or locality parameters.

Figures

Figures reproduced from arXiv: 2605.06182 by Chang-An Zhao, Junjie Huang.

**Figure 1.** Figure 1: The field extensions tower By Proposition 2, we derive that E THAi = Fq(zi) with (zi) E ∞ = |Ai | P|H| u=1 Pu for i = 1, 2. Moreover, we can also derive that, for i = 1, 2, there exist elements e (i) ℓ ∈ E for 2 ≤ ℓ ≤ ri satisfying (7) and e (i) 1 = 1, e (i) 2 , · · · , e (i) ri are linearly independent over E THAi . Define Vi = (X ti j=0 a1,jz j i + Xri ℓ=2 X ti−1 j=0 aℓ,jz j i e (i) ℓ : aℓ,j ∈ Fq ) wi… view at source ↗

read the original abstract

Locally repairable codes with availability have become essential components in modern large-scale distributed cloud storage systems and numerous other applications. In this paper, we focus on the construction of locally repairable codes with one or two recovering sets via elliptic function fields. Prior pioneering work by Li et al. (IEEE Trans. Inf. Theory, vol. 65, no. 1, 2019) and Ma and Xing (J. Comb. Theory Ser. A., vol. 193, 2023) employed maximal supersingular elliptic curves to obtain several optimal (classical) locally repairable codes. In contrast, we consider ordinary elliptic curves with many rational points. This approach yields several new families of \(q\)-ary optimal locally repairable codes with length \(O(q+2\sqrt{q})\) and flexible locality. Consequently, our work broadens the selection of curves available for the construction of optimal locally repairable codes. Furthermore, we present a general framework for constructing locally repairable codes with two recovering sets via automorphism groups of elliptic function fields. To realize this framework, we devise a novel construction for determining the functions \(e_i\) in the construction of locally repairable codes. By employing both supersingular and ordinary elliptic curves, we obtain several families of locally repairable codes with two recovering sets. In particular, we construct a family of \(q^2\)-ary locally repairable codes with two recovering sets, achieving length \(O(q^2+2q)\) and Singleton-defect \(O\!\left(\frac{2\ell}{q^2+2q-8\ell}\right)\), where \(\ell \mid\mid q + 2\) with \(4\ell < q\).

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper shifts LRC constructions to ordinary elliptic curves and adds a general framework for two recovering sets with a new method for the e_i functions.

read the letter

The main point is that this work moves beyond the supersingular elliptic curves in Li et al. and Ma-Xing to ordinary ones. That produces several new q-ary optimal locally repairable code families with lengths on the order of q plus twice the square root of q and flexible locality parameters. They also give a general framework for codes with two recovering sets that works for both curve types, plus a novel way to pick the auxiliary functions e_i from the automorphism group of the elliptic function field. The q-squared-ary family they construct has length roughly q squared plus 2q and a Singleton defect that scales with ell over the length when ell divides q plus 2 and stays below q over 4. These are concrete parameter regimes that could matter for storage systems needing availability. The length and defect bounds rest on standard Hasse point counts, which is solid ground. The constructions are explicit enough to be checked rather than purely existential. One spot that needs scrutiny is the automorphism groups on the ordinary curves. Most ordinary elliptic curves over finite fields have small groups, often just plus or minus the identity. The paper's novel construction for e_i is meant to handle this, but the details will show whether it still produces two disjoint recovering sets without extra conditions that limit the families. If the method works as claimed, the extension is useful; if it only works when the group is large enough, then the ordinary-curve cases shrink. This is for coding theorists who build algebraic codes from curves or function fields. A reader who wants new explicit families with availability would get direct value from the parameter sets and the two-set framework. It is not reshaping the whole field but it does widen the toolkit inside the subarea. I would send it to peer review. The claims are specific, the approach is a clear extension of prior work, and the proofs and constructions are the right things for referees to verify.

Referee Report

2 major / 2 minor

Summary. The paper constructs q-ary optimal locally repairable codes (LRCs) of length O(q + 2√q) with flexible locality by using ordinary elliptic curves with many rational points, and develops a general framework for LRCs with two recovering sets via automorphism groups of elliptic function fields. It introduces a novel method to determine the auxiliary functions e_i and obtains, in particular, a family of q²-ary LRCs with two recovering sets of length O(q² + 2q) and Singleton defect O(2ℓ/(q² + 2q - 8ℓ)) where ℓ divides q + 2 and 4ℓ < q. The work contrasts with prior constructions that relied exclusively on maximal supersingular elliptic curves.

Significance. If the constructions and the novel e_i method are valid, the paper meaningfully broadens the class of curves usable for optimal LRCs by showing that ordinary elliptic curves suffice for both single- and two-recovering-set families, thereby increasing the supply of explicit constructions beyond the supersingular case. The two-recovering-set framework with explicit defect bounds is a concrete advance for availability in distributed storage.

major comments (2)

[two-recovering-set framework] § on two-recovering-set framework: the novel construction of the functions e_i is stated to rely on elements of the automorphism group of the elliptic function field to produce two disjoint recovering sets. For the ordinary curves employed (those achieving approximately q + 2√q rational points), Aut(E) is typically of order at most 6; it is not shown that sufficiently many distinct automorphisms exist to satisfy the disjointness and locality conditions simultaneously for the claimed parameters ℓ || q + 2. Explicit verification or a worked example for at least one such curve is required.
[q-ary families] Abstract and the q-ary families section: the optimality claim for the new q-ary LRCs of length O(q + 2√q) is asserted without an explicit comparison to the LRC Singleton bound or the locality-aware bound used in the paper; the derivation that these codes meet the bound with the stated flexible locality must be supplied or referenced to a precise theorem number.

minor comments (2)

The notation “ℓ || q + 2” should be defined explicitly (presumably meaning ℓ divides q + 2); the same symbol appears in the defect bound without prior clarification.
The O-notation in the length and defect statements should be replaced by explicit asymptotic expressions or concrete upper bounds once the constructions are fully detailed.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the major comments point by point below, indicating where revisions will be made to improve clarity and completeness.

read point-by-point responses

Referee: [two-recovering-set framework] § on two-recovering-set framework: the novel construction of the functions e_i is stated to rely on elements of the automorphism group of the elliptic function field to produce two disjoint recovering sets. For the ordinary curves employed (those achieving approximately q + 2√q rational points), Aut(E) is typically of order at most 6; it is not shown that sufficiently many distinct automorphisms exist to satisfy the disjointness and locality conditions simultaneously for the claimed parameters ℓ || q + 2. Explicit verification or a worked example for at least one such curve is required.

Authors: We appreciate the referee's observation regarding the size of Aut(E) for ordinary elliptic curves. The novel construction of the auxiliary functions e_i is designed to use the (small) automorphism group in a targeted manner: specifically, the group action on the rational points and the associated divisors ensures the two recovering sets are disjoint and satisfy the locality parameter ℓ, with the conditions ℓ | q+2 and 4ℓ < q guaranteeing that the required partitions exist without needing a large group order. The disjointness follows directly from the distinct orbits under the chosen automorphisms and the function field properties. To make this fully explicit as requested, we will add a concrete worked example in the revised manuscript for an ordinary elliptic curve achieving approximately q + 2√q points (e.g., over a small field where the point count is known), listing the automorphisms used, the resulting e_i, and verifying the disjoint recovering sets and locality conditions. revision: partial
Referee: [q-ary families] Abstract and the q-ary families section: the optimality claim for the new q-ary LRCs of length O(q + 2√q) is asserted without an explicit comparison to the LRC Singleton bound or the locality-aware bound used in the paper; the derivation that these codes meet the bound with the stated flexible locality must be supplied or referenced to a precise theorem number.

Authors: We thank the referee for noting this omission. The optimality is with respect to the standard LRC Singleton bound (d ≤ n − k + 1 − ⌈k/r⌉ for locality r, as stated in Theorem 2.3 of the manuscript). In the q-ary families section, the constructed codes from ordinary elliptic function fields achieve equality in this bound for the flexible locality parameters derived from the curve's rational points. In the revision, we will insert an explicit derivation immediately after the code parameter statement, showing that the minimum distance meets the bound with equality, and we will reference the precise theorem number establishing the code parameters. revision: yes

Circularity Check

0 steps flagged

No circularity: novel framework and explicit constructions from elliptic curves

full rationale

The paper cites prior supersingular constructions (Li et al., Ma and Xing) only for context and contrast, then introduces an independent general framework for two-recovering-set LRCs that defines auxiliary functions e_i via a novel method using automorphism groups of elliptic function fields. Both the q-ary optimal LRC families (length O(q+2√q)) and the q²-ary two-recovering-set family (with stated Singleton defect) are obtained by direct construction from ordinary elliptic curves with ~q+2√q points; these curves and their Aut groups are external mathematical objects whose existence is assumed under standard Hasse bounds, not fitted or defined in terms of the target code parameters. No equation reduces a claimed prediction to a fitted input by construction, no uniqueness theorem is imported from self-citation, and the derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard existence results for elliptic curves with many points and on the ability to use their automorphism groups for code construction; no new entities are postulated.

free parameters (2)

q
Finite-field cardinality that determines both the curve and the resulting code length.
ℓ
Divisor parameter satisfying ℓ divides q+2 and 4ℓ < q; appears in the Singleton-defect bound.

axioms (2)

domain assumption Existence of ordinary elliptic curves over F_q possessing approximately q + 2√q rational points
Directly invoked to obtain the stated code lengths O(q+2√q).
domain assumption Automorphism groups of elliptic function fields admit constructions of the auxiliary functions e_i required for two recovering sets
Central to the general framework presented in the abstract.

pith-pipeline@v0.9.0 · 5611 in / 1338 out tokens · 56874 ms · 2026-05-08T04:57:08.147653+00:00 · methodology

discussion (0)

Reference graph

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