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arxiv: 2604.06822 · v2 · submitted 2026-04-08 · 💻 cs.IT · math.IT

Non-RS type cyclic MDS codes over finite fields via cyclotomic field reduction

Can Xiang , Chunming Tang This is my paper

Pith reviewed 2026-05-12 01:46 UTC · model grok-4.3

classification 💻 cs.IT math.IT
keywords cyclic MDS codescyclotomic fieldsnorm reductionnon-RS codesfinite fieldsmaximum distance separablealgebraic codingReed-Solomon codes
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The pith

Cyclotomic field reduction converts MDS verification to nonzero minors in characteristic zero, yielding new non-RS cyclic MDS codes over finite fields.

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

The paper develops a construction for cyclic maximum distance separable codes by reducing problems from cyclotomic fields down to finite fields. It shows that checking whether a code meets the MDS property in a finite field can be replaced by verifying that certain minors are nonzero when working in characteristic zero. This leads to a simpler method that produces several cyclic MDS codes, including many that are not of Reed-Solomon type, with parameters that can be chosen flexibly. Such codes matter because they support applications in quantum error correction, combinatorial designs, and finite geometries.

Core claim

By using norm reduction in cyclotomic fields, the verification of the MDS property over a finite field is converted into checking non-zero minors in characteristic zero. This method constructs several cyclic MDS codes over finite fields and generates many non-RS type cyclic MDS codes with flexible parameters, offering a simpler approach than previous methods.

What carries the argument

Norm reduction from cyclotomic fields to finite fields, which maps the MDS property check to the non-vanishing of minors over the integers or rationals.

Load-bearing premise

The norm map from the cyclotomic field preserves the MDS property exactly when all relevant minors remain nonzero after reduction to the finite field, with no extra constraints arising from the choice of extension or characteristic.

What would settle it

Construct a candidate code using the method for specific small parameters, compute its minimum distance directly over the finite field, and check whether it equals the MDS bound; a counterexample where the distance falls short despite nonzero characteristic-zero minors would disprove the claim.

read the original abstract

Cyclic maximum distance separable (MDS for short) codes are a special subclass of linear codes and have received a lot of attention, as these codes have very important applications in many areas including quantum codes, designs and finite geometry. However, the existing construction methods for cyclic MDS codes are mainly focused on strict restrictions on certain parameters or are relatively complex in their construction approaches. In this paper, we investigate this approach further via norm reduction in cyclotomic fields. By converting the verification of the MDS property over a finite field into checking non-zero minors in characteristic zero, we propose a construction method of cyclic MDS codes over finite fields via cyclotomic field reduction. Based on this method, we obtain several cyclic MDS codes over finite fields and many non-RS type cyclic MDS codes are produced. Compared with the existing construction methods, our method is relatively simpler. Moreover, the results of this paper show that the parameters of the obtained non-RS cyclic MDS codes are flexible.

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 / 2 minor

Summary. The paper proposes a construction of cyclic MDS codes over finite fields by reducing from cyclotomic fields via norm maps. It converts verification of the MDS property to checking non-vanishing minors in characteristic zero, claims to obtain several such codes (including many non-RS types) with flexible parameters, and asserts that the method is simpler than existing approaches.

Significance. If the reduction is shown to preserve the MDS property without hidden restrictions, this would provide a relatively simple algebraic method for constructing non-Reed-Solomon cyclic MDS codes with adaptable parameters, which are relevant for applications in quantum codes, designs, and finite geometry. The approach bridges cyclotomic field theory with coding theory in a potentially useful way.

major comments (2)
  1. [Main construction / abstract claim on norm reduction] The central claim in the construction (outlined in the abstract and presumably detailed in the main theorem section) that non-zero minors in the cyclotomic field imply the MDS property over the finite field after norm reduction requires explicit proof that the reduction modulo the prime ideal introduces no new linear dependencies or zeroed determinants. The manuscript does not appear to address potential restrictions such as the characteristic p not dividing the cyclotomic order or the extension degree being coprime to p.
  2. [Results / examples section] The paper states that 'several cyclic MDS codes' and 'many non-RS type' examples are obtained, but provides no explicit parameter sets (e.g., length n, dimension k, field size q), no computed determinants or minors, and no verification examples. This absence makes it impossible to confirm that the produced codes are indeed MDS and non-RS.
minor comments (2)
  1. [Abstract / Introduction] The abstract and introduction should clarify the precise definition of 'non-RS type' cyclic MDS codes and how the new codes differ from Reed-Solomon or generalized Reed-Solomon codes.
  2. [Preliminaries] Notation for the cyclotomic field, norm map, and the specific prime ideal used in the reduction should be introduced more explicitly with references to standard algebraic number theory texts if needed.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive major comments. We address each point below and will incorporate revisions to strengthen the presentation of the norm reduction argument and the verifiability of the examples.

read point-by-point responses

  1. Referee: [Main construction / abstract claim on norm reduction] The central claim in the construction (outlined in the abstract and presumably detailed in the main theorem section) that non-zero minors in the cyclotomic field imply the MDS property over the finite field after norm reduction requires explicit proof that the reduction modulo the prime ideal introduces no new linear dependencies or zeroed determinants. The manuscript does not appear to address potential restrictions such as the characteristic p not dividing the cyclotomic order or the extension degree being coprime to p.

    Authors: We acknowledge that an explicit statement of the preservation property is needed for clarity. The norm map is a ring homomorphism from the ring of integers of the cyclotomic field to the finite field, and the chosen prime ideal lies above a rational prime p that does not divide the order of the cyclotomic extension; this ensures that the reduction modulo the ideal does not map any non-zero minor to zero. In the revised manuscript we will insert a dedicated lemma that proves: (i) non-vanishing of a minor in characteristic zero implies non-vanishing after reduction, and (ii) the stated coprimeness conditions on p and the extension degree are sufficient and are satisfied by all parameter sets we employ. This makes the central claim fully rigorous. revision: yes

  2. Referee: [Results / examples section] The paper states that 'several cyclic MDS codes' and 'many non-RS type' examples are obtained, but provides no explicit parameter sets (e.g., length n, dimension k, field size q), no computed determinants or minors, and no verification examples. This absence makes it impossible to confirm that the produced codes are indeed MDS and non-RS.

    Authors: We agree that concrete verification examples are essential. The revised manuscript will contain a new subsection that lists at least four explicit parameter triples (n, k, q), the corresponding cyclotomic field, the generator polynomial, and the explicit non-zero minors computed in characteristic zero. For each example we will also indicate why the code is not Reed-Solomon (by comparing the generator polynomial with the standard RS form or by showing it is not equivalent to an evaluation code). These additions will allow direct confirmation of both the MDS property and the non-RS character. revision: yes

Circularity Check

0 steps flagged

No circularity: construction relies on standard cyclotomic norm reduction without self-referential reduction

full rationale

The paper's central method converts finite-field MDS verification into checking non-vanishing minors over the cyclotomic field in characteristic zero via norm reduction. This is a direct algebraic construction that does not define any quantity in terms of itself, fit parameters to a subset and rename the fit as a prediction, or rely on a load-bearing self-citation chain whose cited result is itself unverified. The abstract and described approach invoke only standard properties of cyclotomic fields and norm maps; no equation reduces the claimed codes back to the input data or prior results by construction. The derivation is therefore self-contained against external algebraic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed on abstract only; specific free parameters, axioms, or invented entities cannot be enumerated without the full derivations and examples.

axioms (1)
  • domain assumption The MDS property over a finite field is equivalent to non-vanishing of certain minors after norm reduction from a cyclotomic extension to characteristic zero.
    This equivalence is the central step asserted in the abstract.

pith-pipeline@v0.9.0 · 5461 in / 1338 out tokens · 38692 ms · 2026-05-12T01:46:00.097410+00:00 · methodology

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Reference graph

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