REVIEW 2 major objections 2 minor 3 references
Locally recoverable codes from elliptic surfaces with availability and hierarchical locality
T0 review · 2 major / 2 minor · reviewed 2026-06-29 · grok-4.3
Pith's one-line read Constructions from elliptic surfaces produce locally recoverable codes with availability t greater than 2 and hierarchical locality.
desk verdict The paper gives new elliptic-surface constructions for LRCs that combine t>2 availability with hierarchical locality, but the explicit translation from fibration and torsion to recovery sets and parameters needs checking. 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 fibered structure of elliptic surfaces and the torsion groups of the elliptic curves, which together create a multi-dimensional setting for nested recovery sets in the codes.
What would settle it
An explicit example of an elliptic surface where the resulting code does not exhibit the claimed availability or hierarchical locality would falsify the constructions.
Extended reading notes
Core claim
Several constructions of locally recoverable codes are proposed from elliptic surfaces. These yield codes with availability t>2, codes with hierarchical locality, and codes combining both. The constructions rely on the properties of the torsion groups of elliptic curves and on the fibered structure of elliptic surfaces to introduce a multi-dimensional setting that allows for more recovery sets, eventually nested one within another.
Load-bearing premise
The geometry of the surface and the torsion groups of the elliptic curves suffice to introduce a multi-dimensional setting that produces the desired nested recovery sets.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proposes several constructions of locally recoverable codes (LRCs) from elliptic surfaces. It claims to obtain codes with availability t>2, codes with hierarchical locality, and codes combining both properties. The constructions rely on the torsion groups of the elliptic curves and the fibered structure of the surfaces to introduce a multi-dimensional setting that produces the desired (possibly nested) recovery sets.
Significance. If the geometric constructions map explicitly to verifiable LRC parameters with the stated locality and availability properties, the work would provide new algebraic-geometry-based methods for designing LRCs with advanced features such as t>2 availability and hierarchical locality. Such constructions could be of interest for distributed storage applications, provided they yield competitive (n,k,r) or (n,k,r,t) tuples and explicit recovery functions.
major comments (2)
- [§4] §4 (Constructions from elliptic surfaces): The central claim that the fibration and torsion subgroups induce explicit nested recovery sets for hierarchical locality (and t>2 availability) is asserted via the 'geometry of the surface' and 'torsion groups,' but the manuscript does not supply an explicit definition of the recovery functions, a verification that each symbol is a deterministic function of its recovery set, or a parameter count showing the resulting locality parameters. This mapping is load-bearing for all three claimed families of codes.
- [§5] §5 (Parameter tables or examples): No explicit (n,k,r) or (n,k,r,t) tuples, minimum-distance bounds, or rate calculations are provided that would allow verification that the geometric construction produces nontrivial locality; without these, the coding-theoretic claims cannot be assessed against existing LRC literature.
minor comments (2)
- [Introduction] The abstract and introduction use the phrase 'multi-dimensional setting' without a preliminary definition or diagram; a short subsection clarifying the coordinate system induced by the fibration would improve readability.
- [Preliminaries] Notation for the recovery sets (e.g., how the torsion points label the sets) should be introduced consistently before the constructions are stated.
Simulated Author's Rebuttal
We thank the referee for the careful reading of our manuscript and the constructive comments. We address the two major comments below and will revise the manuscript to incorporate the requested clarifications and examples.
read point-by-point responses
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Referee: [§4] §4 (Constructions from elliptic surfaces): The central claim that the fibration and torsion subgroups induce explicit nested recovery sets for hierarchical locality (and t>2 availability) is asserted via the 'geometry of the surface' and 'torsion groups,' but the manuscript does not supply an explicit definition of the recovery functions, a verification that each symbol is a deterministic function of its recovery set, or a parameter count showing the resulting locality parameters. This mapping is load-bearing for all three claimed families of codes.
Authors: We agree that explicit definitions and verifications are necessary for the claims to be fully verifiable. In the revised manuscript we will add explicit definitions of the recovery functions derived from the torsion subgroups of the fibers and the fibration structure of the surface, include a direct verification that each symbol is a deterministic function of its recovery set(s), and provide the corresponding locality parameter counts for the three families of constructions. revision: yes
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Referee: [§5] §5 (Parameter tables or examples): No explicit (n,k,r) or (n,k,r,t) tuples, minimum-distance bounds, or rate calculations are provided that would allow verification that the geometric construction produces nontrivial locality; without these, the coding-theoretic claims cannot be assessed against existing LRC literature.
Authors: We acknowledge that concrete parameter examples are required to allow assessment against the existing literature. In the revision we will include a new subsection with explicit (n,k,r) and (n,k,r,t) tuples, minimum-distance bounds, and rate calculations for representative constructions, together with a brief comparison to known LRC families. revision: yes
Circularity Check
No circularity; constructions derive from external elliptic geometry and torsion groups.
full rationale
The paper's claims rest on standard properties of elliptic surfaces and their torsion subgroups to define multi-dimensional recovery sets. No equations or definitions are shown to reduce to self-reference, fitted parameters renamed as predictions, or load-bearing self-citations. The abstract explicitly ties the nested recovery sets to the fibration structure and torsion, which are independent mathematical inputs. This matches the default expectation that most papers are non-circular; the derivation chain remains self-contained against external geometric facts.
Assumptions & free parameters
Cite this review
Pith. "Pith review of Locally recoverable codes from elliptic surfaces with availability and hierarchical locality." pith.science (2026). https://pith.science/paper/SRKRFUAG
@misc{pith2026260528460,
author = {Pith},
title = {Pith review of: Locally recoverable codes from elliptic surfaces with availability and hierarchical locality},
year = {2026},
howpublished = {\url{https://pith.science/paper/SRKRFUAG}},
note = {Machine review of arXiv:2605.28460}
}
abstract
In this paper, we propose several constructions of Locally Recoverable Codes from elliptic surfaces. In particular, we are able to obtain codes with availability $t>2$, codes with hierarchical locality and, finally, codes which combine availability and hierarchical locality. Our constructions rely on the properties of the torsion groups of elliptic curves and on the fibered structure of elliptic surfaces. In particular, the geometry of the surface is used to introduce a multi-dimensional setting, allowing for more recovery sets, eventually nested one within another.
Reference graph
Works this paper leans on
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[1]
Ruled surfaces over finite fields, and some codes over them
[BH25] Régis Blache and Emmanuel Hallouin. Ruled surfaces over finite fields, and some codes over them. arXiv preprint arXiv:2509.18698,
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[2]
Locally recoverable codes from algebraic curves and surfaces
[BHH+17] Alexander Barg, Kathryn Haymaker, Everett W Howe, Gretchen L Matthews, and Anthony Várilly- Alvarado. Locally recoverable codes from algebraic curves and surfaces. InAlgebraic Geometry for Coding Theory and Cryptography: IPAM, Los Angeles, CA, February 2016, pages 95–127. Springer,
2016
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[3]
[SV25] Cecília Salgado and Lara Vicino. Locally recoverable codes with availability from a family of fibered surfaces.arXiv preprint arXiv:2512.08100,
Reviewed June 29, 2026 · model on record in the stance chip above.
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