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arxiv: 2606.09522 · v1 · pith:W7LK7TXKnew · submitted 2026-06-08 · 💻 cs.IT · math.IT

Constructions of Quantum (r,δ)-LRCs from cyclic codes

Rajendra Prasad Rajpurohit , Maheshanand Bhaintwal This is my paper

Pith reviewed 2026-06-27 14:45 UTC · model grok-4.3

classification 💻 cs.IT math.IT
keywords quantum locally recoverable codescyclic codesCSS construction(r,δ)-LRCsquantum Singleton bounddual-containing codesfinite fieldserasure recovery
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The pith

Cyclic classical (r,δ)-LRCs whose defining sets satisfy a dual-containing condition produce three families of quantum (r,δ)-LRCs via CSS, with two families optimal under the quantum Singleton-like bound when pure.

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 obtain quantum locally recoverable codes by selecting classical cyclic (r,δ)-LRCs whose defining sets meet the dual-containing condition needed for the CSS construction. Three explicit families are built this way. Two of the families meet the quantum Singleton-like bound whenever the resulting quantum codes are pure. Constructions 2 and 3 impose no upper bound on code length relative to the size of the finite field.

Core claim

We present three explicit families of (r,δ)-qLRCs constructed from cyclic classical (r,δ)-LRCs via the CSS method, where the defining sets satisfy the dual-containing condition. Two of these families attain the quantum Singleton-like bound when the codes are pure, and constructions 2 and 3 allow code lengths unbounded by the field size.

What carries the argument

The dual-containing condition on the defining sets of cyclic (r,δ)-LRCs that permits the CSS construction to produce quantum (r,δ)-LRCs.

If this is right

  • The constructions supply explicit quantum codes that recover from erasures by reading only a small number of other symbols.
  • Optimal quantum LRC examples exist for the Singleton-like bound in the pure case.
  • Codes from constructions 2 and 3 can be realized at any desired length over a field of any size.
  • The dual-containing cyclic LRCs serve as systematic building blocks for quantum codes with locality.

Where Pith is reading between the lines

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

  • The absence of a length-to-field-size bound may allow these codes to scale to larger quantum storage systems than field-size-restricted constructions.
  • The same dual-containing selection technique could be tested on non-cyclic classical LRCs to produce additional quantum families.
  • Purity of the quantum codes may be verified directly from the classical defining sets for the optimal cases.
  • The approach connects classical distributed-storage techniques to quantum error correction without requiring new algebraic structures beyond cyclic codes.

Load-bearing premise

The selected cyclic (r,δ)-LRCs have defining sets that satisfy the dual-containing condition required for the CSS construction, and the quantum codes are pure when optimality is claimed.

What would settle it

For a concrete parameter triple (r,δ,q) in one of the three families, compute the actual minimum distance and locality parameters of the resulting quantum code and check whether they match the claimed values and satisfy the quantum Singleton-like bound in the pure case.

read the original abstract

Classical $(r,\delta)$ locally recoverable codes (LRCs) play a central role in distributed data storage systems as they enable an efficient recovery from erasures by accessing a small number of surviving symbols. Motivated by their prospective use in future quantum data storage and by recent theoretical progress on quantum locally recoverable codes (qLRCs), we investigate the construction of qLRCs from classical cyclic $(r,\delta)$-LRCs. Our approach identifies cyclic LRCs whose defining sets satisfy a dual-containing condition, allowing them to serve as valid CSS ingredients. We present three explicit families of $(r,\delta)$-qLRCs, two of which are optimal with respect to the quantum Singleton-like bound, whenever the codes are pure, thereby providing optimal examples. Additionally, the codes presented in Constructions 2 and 3 have no bound on their lengths with respect to the field size required to obtain these 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.

Referee Report

2 major / 1 minor

Summary. The manuscript constructs three explicit families of quantum (r,δ)-LRCs from classical cyclic (r,δ)-LRCs whose defining sets satisfy the dual-containing condition, allowing application of the CSS construction. Two families are asserted to meet the quantum Singleton-like bound whenever the resulting codes are pure, with Constructions 2 and 3 having no length restriction relative to the field size.

Significance. If the constructions are explicit and purity can be established, the results supply concrete optimal qLRC examples without length-field-size bounds, which would be useful for quantum storage applications and extend known CSS-based qLRC constructions.

major comments (2)
  1. [Constructions 2 and 3] Constructions 2 and 3: optimality with respect to the quantum Singleton-like bound is claimed only 'whenever the codes are pure,' yet no argument, weight enumeration, or check is supplied showing that the minimum distance of the CSS code equals the designed distance (i.e., that no linear combination of parity-check rows yields a nonzero vector of weight less than d with support inside the information set). This verification is load-bearing for the 'optimal examples' assertion.
  2. [Defining sets and dual-containing condition] Defining-set selection for dual containment: while the condition C⊥ ⊆ C is used to obtain valid CSS ingredients, the manuscript does not derive or bound the actual minimum distance of the quantum code beyond the designed value, leaving the purity-dependent optimality claim without supporting evidence.
minor comments (1)
  1. [Abstract] Abstract: the phrase 'whenever the codes are pure' is stated without indicating for which parameter regimes purity holds or supplying even one concrete parameter set where purity is verified.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments. We respond to each major comment below.

read point-by-point responses

  1. Referee: [Constructions 2 and 3] Constructions 2 and 3: optimality with respect to the quantum Singleton-like bound is claimed only 'whenever the codes are pure,' yet no argument, weight enumeration, or check is supplied showing that the minimum distance of the CSS code equals the designed distance (i.e., that no linear combination of parity-check rows yields a nonzero vector of weight less than d with support inside the information set). This verification is load-bearing for the 'optimal examples' assertion.

    Authors: The manuscript states the optimality claim conditionally on the resulting codes being pure, without providing a general argument, weight enumeration, or verification that the actual minimum distance equals the designed distance for all parameters. The constructions are explicit, so purity and distance can be checked for concrete parameter choices, but no such general supporting evidence is included. We will revise the manuscript to clarify the conditional nature of the claim and add discussion or examples illustrating cases where purity holds. revision: yes

  2. Referee: [Defining sets and dual-containing condition] Defining-set selection for dual containment: while the condition C⊥ ⊆ C is used to obtain valid CSS ingredients, the manuscript does not derive or bound the actual minimum distance of the quantum code beyond the designed value, leaving the purity-dependent optimality claim without supporting evidence.

    Authors: The dual-containing condition on the defining sets is used to ensure the CSS construction is applicable, yielding a quantum code whose distance is at least the designed value. The optimality statement is explicitly conditioned on purity (i.e., equality to the designed distance). We agree that the manuscript does not derive a general bound on the actual distance or conditions guaranteeing purity, which leaves the claim without additional supporting evidence beyond the conditional statement. We will revise to include further clarification on this point. revision: yes

Circularity Check

0 steps flagged

No circularity: explicit constructions from defining-set conditions

full rationale

The paper derives three explicit families of (r,δ)-qLRCs by selecting cyclic classical LRCs whose defining sets satisfy the dual-containing condition C⊥ ⊆ C, then applying the CSS construction. Optimality is asserted only conditionally on purity with respect to the quantum Singleton-like bound; this is a standard external verification criterion in quantum coding theory and does not reduce any claimed parameter to a quantity fitted or defined inside the paper. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear in the derivation chain. The constructions are self-contained against the stated algebraic conditions on cyclic codes.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The constructions rest on standard algebraic properties of cyclic codes and the CSS framework; no new entities or fitted parameters are introduced in the abstract.

axioms (2)
  • standard math Cyclic codes are completely determined by their defining sets consisting of cyclotomic cosets.
    Standard fact from finite-field algebra used to describe the classical LRCs.
  • domain assumption A classical linear code whose defining set satisfies the dual-containing condition yields a valid CSS quantum code.
    Invoked directly in the abstract to convert classical cyclic LRCs into qLRCs.

pith-pipeline@v0.9.1-grok · 5695 in / 1171 out tokens · 22853 ms · 2026-06-27T14:45:34.192360+00:00 · methodology

discussion (0)

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