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Extending the Chen-Ding construction to even ord_n(q) produces self-dual cyclic codes whose minimum distances satisfy square-root lower bounds.

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2026-07-01 03:39 UTC pith:G3FDWK4U

load-bearing objection Extends Chen-Ding to even ord_n(q) with square-root bounds via consecutive zeros and a refined parameter pick that improves distances in the other cases.

arxiv 2606.31294 v1 pith:G3FDWK4U submitted 2026-06-30 cs.IT math.IT

Self-Dual Cyclic Codes with Improved Minimum Distance Estimates via Extending the Chen-Ding Construction

Bofeng Huang , Jingwei Zhang , Chang-An Zhao This is my paper
classification cs.IT math.IT
keywords self-dual cyclic codesminimum distance boundsChen-Ding constructionEuclidean self-dual codesHermitian self-dual codesdefining setcyclic codes over finite fields
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Chen and Ding constructed self-dual cyclic codes when the multiplicative order ord_n(q) is odd. This work extends the construction to even orders and shows that the resulting codes meet square-root lower bounds on minimum distance. By studying consecutive zero segments in the defining set of the dual code, the authors obtain exact parameters for Euclidean self-dual cyclic codes with even ord_n(q) and for Hermitian self-dual cyclic codes with odd ord_n(q). They further introduce refined parameter choices that increase the minimum distance for the remaining families while preserving length and dimension.

Core claim

Extending the Chen-Ding construction to the case of even ord_n(q) yields self-dual cyclic codes whose minimum distances satisfy square-root lower bounds; examining consecutive zero segments in the defining set of the dual code determines the exact parameters of Euclidean self-dual cyclic codes with even ord_n(q) and Hermitian self-dual cyclic codes with odd ord_n(q), while refined parameter selection produces larger minimum distances for Euclidean self-dual codes with odd ord_n(q) and Hermitian self-dual codes with even ord_n(q).

What carries the argument

Consecutive zero segments in the defining set of the dual code, used to derive square-root lower bounds and exact parameters.

Load-bearing premise

Examining consecutive zero segments in the defining set of the dual code is enough to prove the square-root lower bounds and exact parameters when ord_n(q) is even.

What would settle it

An explicit Euclidean self-dual cyclic code with even ord_n(q) whose minimum distance is strictly less than the square root of its length would disprove the lower-bound claim.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Self-dual cyclic codes with even ord_n(q) achieve minimum distance at least the square root of the code length.
  • Exact parameters become known for Euclidean self-dual codes with even ord_n(q) and Hermitian self-dual codes with odd ord_n(q).
  • Refined parameter selection yields strictly larger minimum distances for the remaining Euclidean and Hermitian families at fixed length and dimension.
  • Several existing families of self-dual cyclic codes receive improved lower bounds on minimum distance.

Where Pith is reading between the lines

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

  • The segment-analysis technique may extend to other families of cyclic or constacyclic codes whose defining sets contain long runs of zeros.
  • The refined constructions could be tested numerically for small even orders to measure how often the new distance exceeds the basic square-root bound.
  • Applications that use self-dual codes for quantum error correction or lattice constructions may benefit from the larger minimum distances obtained by the refined choices.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 2 minor

Summary. The paper extends the Chen-Ding construction of self-dual cyclic codes from the odd ord_n(q) case to even ord_n(q). It establishes square-root lower bounds on minimum distance for the resulting codes by analyzing consecutive zero segments in the defining set of the dual code. Exact parameters are determined for Euclidean self-dual cyclic codes with even ord_n(q) and Hermitian self-dual cyclic codes with odd ord_n(q). For Euclidean self-dual codes with odd ord_n(q) and Hermitian self-dual codes with even ord_n(q), a refined parameter selection is introduced that yields larger minimum distances at fixed length and dimension, along with tighter lower bounds for several families of such codes.

Significance. If the derivations hold, the work provides a systematic extension of known constructions and a concrete method (consecutive zero segments) for obtaining exact parameters and square-root bounds in additional cases. This enriches the algebraic coding theory of self-dual cyclic codes and supplies improved distance estimates that may be useful for applications requiring high-distance self-dual codes.

minor comments (2)
  1. [Abstract] Abstract: the phrase 'refined parameter selection' is used without indicating the precise criterion or optimization used; a brief parenthetical description would clarify the contribution.
  2. The manuscript would benefit from an explicit comparison table (new bounds versus prior Chen-Ding bounds and BCH bounds) for at least one family to quantify the improvement claimed in the final paragraph.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive evaluation of our manuscript and the recommendation of minor revision. The referee's summary correctly reflects the main contributions: the extension of the Chen-Ding construction to even ord_n(q), the square-root bounds via consecutive zero segments, the exact parameter determinations in selected cases, and the refined parameter choices yielding improved distances. No major comments requiring technical response were provided.

Circularity Check

0 steps flagged

No significant circularity; derivation applies standard BCH bound to extended construction

full rationale

The paper extends the Chen-Ding construction (distinct prior authors) to even ord_n(q) by examining consecutive zero segments in the dual defining set to obtain square-root lower bounds and exact parameters via the BCH bound. This is a direct algebraic application of the standard consecutive-zeros lower bound combined with the self-duality closure condition; no equations reduce to fitted inputs, no self-citations are load-bearing for the central claim, and no ansatz or uniqueness theorem is smuggled in. The derivation is self-contained against external coding-theory benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Report based solely on abstract; no free parameters, axioms, or invented entities are identifiable from the provided text.

pith-pipeline@v0.9.1-grok · 5787 in / 1126 out tokens · 35286 ms · 2026-07-01T03:39:00.545086+00:00 · methodology

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read the original abstract

Self-dual cyclic codes have garnered significant interest owing to their rich algebraic structures and wide-ranging applicability. Their construction and the establishment of lower bounds on their minimum distances are fundamental problems in coding theory. Chen and Ding laid an important foundation for the construction of self-dual cyclic codes in the case where the multiplicative order of $q$ module $n$, denoted by $\operatorname{ord}_n(q)$, is odd. Building on their work, we extend the investigation to the case of even order $\operatorname{ord}_n(q)$ and demonstrate that the minimum distances of the resulting self-dual cyclic codes satisfy square-root lower bounds. By examining the consecutive zero segments in the defining set of the dual code, we determine the exact parameters of Euclidean self-dual cyclic codes with even $\operatorname{ord}_n(q)$ and Hermitian self-dual cyclic codes with odd $\operatorname{ord}_n(q)$. Furthermore, for Euclidean self-dual cyclic codes with odd $\operatorname{ord}_n(q)$ and Hermitian self-dual cyclic codes with even $\operatorname{ord}_n(q)$, we introduce a refined parameter selection that leads to larger minimum distances with the same code length and dimension. This approach also yields tighter lower bounds for several families of self-dual cyclic codes. This work enriches the theory of self-dual cyclic codes and offers new insights into estimating lower bounds on their minimum distances.

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Reference graph

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