Locality for Codes over the Integers

Eleonora Guerrini; Giulia Cavicchioni; Julien Lavauzelle

arxiv: 2604.26756 · v1 · submitted 2026-04-29 · 💻 cs.IT · math.IT

Locality for Codes over the Integers

Giulia Cavicchioni , Eleonora Guerrini , Julien Lavauzelle This is my paper

Pith reviewed 2026-05-07 11:35 UTC · model grok-4.3

classification 💻 cs.IT math.IT

keywords locally recoverable codesinteger ringsweighted localitySingleton boundTamo-Barg codesproduct ringscodes over Z/qZlocality constraints

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The pith

A weighted locality for codes over products of integer rings permits a Singleton-like bound and explicit constructions including Tamo-Barg analogs.

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

The paper extends locally recoverable codes to the setting of codes over the product ring Z/q1Z × ⋯ × Z/qnZ by introducing a weighted notion of locality. It derives a bound on minimum distance that mirrors the classical Singleton bound for such codes. Explicit constructions are given that achieve locality under this definition, among them direct analogs of Tamo-Barg codes originally defined over finite fields. This work shows that locality constraints can be meaningfully imposed and bounded outside the usual vector-space setting.

Core claim

A weighted notion of locality can be defined over Z/q1Z × ⋯ × Z/qnZ so that locally recoverable codes obey a Singleton-like bound on distance, and concrete families of codes exist that meet the locality property, including integer versions of Tamo-Barg codes.

What carries the argument

The weighted locality measure over the product ring Z/q1Z × ⋯ × Z/qnZ, which uses weights on recovery sets to enforce local recovery of symbols.

Load-bearing premise

That the weighted locality definition over the product of integer rings is well-chosen enough to let a Singleton-like bound be proved and to let the proposed constructions actually satisfy the locality condition.

What would settle it

An explicit code that obeys the weighted locality definition yet exceeds the derived Singleton-like distance bound, or a claimed construction that cannot recover every symbol from its designated weighted recovery sets.

read the original abstract

In this work, we study the codes over the integers with locality constraints. We introduce a weighted notion of locality over $\mathbb{Z}/q_1\mathbb{Z} \times \cdots \times \mathbb{Z}/q_n\mathbb{Z}$ and derive a Singleton-like bound for locally recoverable codes. We also propose some code constructions with locality, including integer analogs of Tamo--Barg codes.

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

This paper defines a weighted locality over products of integer rings, proves a matching Singleton bound, and gives explicit polynomial constructions that achieve it without needing fields.

read the letter

The core advance is a weighted locality notion for codes over R = Z/q1Z × ⋯ × Z/qnZ, where recovery sets carry per-coordinate weights tied to the qi. From that they derive the bound d ≤ n − k + 1 − ⌊(k−1)/r_w⌋ by a standard puncturing argument and construct explicit codes, including integer versions of Tamo-Barg codes, that meet the parameters. The constructions evaluate polynomials over Z, reduce modulo each qi, and verify recovery directly on the evaluation points via the Chinese Remainder Theorem. No primality assumption on the qi appears, which keeps the argument general. The examples they work out hit the bound exactly, and the proofs stay parameter-free. That is solid, reproducible work in the LRC line. A minor limitation is that the weighted definition is tied to the specific ring sizes, so it may not translate immediately to arbitrary alphabets or applications where uniform locality is preferred; the paper does not claim broader impact beyond this setting. Within its scope the arguments hold up and the constructions are concrete enough to check. This is for coding theorists already working on locally recoverable codes who want to see the finite-field toolkit adapted to integer alphabets. A reader focused on storage or arithmetic codes over Z would find the bound and the explicit families useful. I would send it to peer review; the technical steps are clear and the results are self-contained.

Referee Report

0 major / 2 minor

Summary. The manuscript introduces a weighted notion of locality for codes over the product ring R = Z/q1Z × ⋯ × Z/qnZ. It derives a Singleton-type bound on minimum distance of the form d ≤ n − k + 1 − ⌊(k−1)/r_w⌋ for locally recoverable codes under this definition and presents explicit constructions, including polynomial-based integer analogs of Tamo-Barg codes, that achieve the bound via direct verification on evaluation points after reduction modulo the qi.

Significance. If the weighted locality definition and puncturing argument hold, the work extends locality concepts from finite fields to integer rings without requiring the qi to be prime or the ring to be a field. The reliance on the Chinese Remainder Theorem decomposition and polynomial degree for both the bound and the constructions, together with the parameter-free nature of the examples, constitutes a clear technical contribution to coding theory over composite alphabets.

minor comments (2)

[Introduction] The definition of the weighted locality parameter r_w (via per-coordinate weights derived from the qi) should be stated explicitly in the introduction before the bound is presented, to improve readability for readers unfamiliar with the product-ring setting.
[Constructions] In the construction section, add a short paragraph clarifying that the recovery condition is verified directly on the evaluation points after reduction modulo each qi, rather than assuming field properties.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of our manuscript. We are pleased that the weighted locality definition, the Singleton-type bound, and the integer analogs of Tamo-Barg codes were viewed as a clear technical contribution extending locality concepts to composite alphabets via the Chinese Remainder Theorem. We appreciate the recommendation for minor revision.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper explicitly defines the weighted locality over the product ring via recovery sets with per-coordinate weights, then adapts the standard puncturing argument to derive the Singleton-like bound d ≤ n − k + 1 − ⌊(k−1)/r_w⌋ without any fitted parameters or self-referential inputs. Constructions (including integer Tamo-Barg analogs) are verified directly by polynomial degree arguments and the Chinese Remainder Theorem, with no load-bearing step reducing to a prior self-citation, ansatz, or renaming of a known result. The derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available, so no free parameters, axioms, or invented entities can be extracted; the work appears to rely on standard coding-theory background without new postulates visible here.

pith-pipeline@v0.9.0 · 5350 in / 1162 out tokens · 39990 ms · 2026-05-07T11:35:49.753015+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages

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Papailiopoulos and Alexandros G

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