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arxiv: 2605.23329 · v1 · pith:R66HMHGOnew · submitted 2026-05-22 · 💻 cs.IT · math.IT

MDS and NMDS Codes from the Extended Twisted Generalized Reed-Solomon Codes

Yanli Wang , Yanxin Chen , Tongjiang Yan This is my paper

Pith reviewed 2026-05-25 03:07 UTC · model grok-4.3

classification 💻 cs.IT math.IT
keywords MDS codesNMDS codesextended TGRS codesSchur productdual codesgenerator matrixReed-Solomon codesAMDS codes
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The pith

Appending three columns to the generator matrix of twisted generalized Reed-Solomon codes yields MDS or NMDS codes under derived conditions.

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

The paper builds a family of extended TGRS codes by adding three specific columns to the generator matrix of existing TGRS codes. It then derives the necessary and sufficient conditions on the parameters that make these new codes MDS or almost MDS. Analysis of the dual codes supplies the corresponding conditions for the codes to be NMDS. Examples are provided to check the conditions, and the Schur product is used to prove the resulting codes are not generalized Reed-Solomon codes. Such constructions matter because MDS and NMDS codes achieve the best possible error-correction capability for their length and dimension.

Core claim

The paper constructs extended TGRS codes by appending three columns to the generator matrix of TGRS codes. It derives necessary and sufficient conditions for these codes to be MDS or AMDS, establishes necessary and sufficient conditions for them to be NMDS by checking the AMDS property of their duals, and proves they are non-GRS using the Schur product method.

What carries the argument

The extended TGRS codes obtained by appending three columns to the TGRS generator matrix, whose minimum distance is controlled by the interaction of those columns with the original rows.

If this is right

  • New infinite families of MDS codes arise whenever the column-appending conditions hold.
  • The dual analysis directly produces NMDS codes from the same construction.
  • The Schur-product test separates these codes from all generalized Reed-Solomon codes.
  • Concrete parameter sets can be checked against the conditions to produce explicit optimal codes.

Where Pith is reading between the lines

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

  • The same column-extension technique could be tested on other twisted or generalized Reed-Solomon variants to generate further MDS examples.
  • If the non-GRS property persists under field extensions, these codes might offer different weight distributions useful for applications beyond the Singleton bound.
  • The conditions could be specialized to small finite fields to produce tables of short optimal codes for immediate implementation checks.

Load-bearing premise

The three appended columns must be chosen so that they preserve the required rank and distance properties of the original TGRS matrix under the stated parameter restrictions.

What would settle it

An explicit choice of parameters satisfying the stated conditions for which the minimum distance of the constructed code is strictly less than the MDS or NMDS bound.

read the original abstract

This paper contributes to maximum distance separable (MDS) and near MDS (NMDS) properties of the extended generalized twisted Reed-Solomon (TGRS) codes. Firstly, a family of extended TGRS (ETGRS) are constructed by appending three columns to the generator matrix of original TGRS codes. Secondly, the necessary and sufficient conditions for these codes to be MDS or almost MDS (AMDS) codes are derived. Then, by analyzing the AMDS properties of their dual codes, the necessary and sufffcient conditions for them to be NMDS codes are established. Furthermore, some examples are given to verify the main results. Finally, we determine the non-generalized Reed-Solomon (non-GRS) characteristics of them via the Schur product method.

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

0 major / 2 minor

Summary. The paper constructs a family of extended twisted generalized Reed-Solomon (ETGRS) codes by appending three columns to the generator matrix of original TGRS codes. It derives necessary and sufficient conditions for these ETGRS codes to be MDS or almost MDS (AMDS). Necessary and sufficient conditions for NMDS codes are then obtained by analyzing the AMDS properties of the dual codes. Examples are provided to verify the results, and the non-GRS character of the codes is established via the Schur product method.

Significance. If the stated conditions hold under the given parameter restrictions, the work supplies explicit new constructions of MDS, AMDS, and NMDS codes extending the TGRS family. Such constructions are of interest in coding theory because MDS and NMDS codes achieve or approach the Singleton bound and have applications in distributed storage and network coding. The Schur-product verification of the non-GRS property follows a standard technique in the literature and strengthens the claim that these are genuinely new codes rather than disguised GRS codes.

minor comments (2)
  1. [Abstract] Abstract: 'sufffcient' is a typographical error and should read 'sufficient'.
  2. [Abstract] The abstract states that necessary and sufficient conditions are derived but does not indicate the precise ranges of the twisting parameters or the field characteristic under which the three appended columns preserve the required minimum distance; a brief statement of these ranges would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our work on extended TGRS codes and for recommending minor revision. The referee's description accurately reflects the manuscript's contributions regarding MDS/AMDS/NMDS conditions and the non-GRS verification.

Circularity Check

0 steps flagged

No circularity; derivation is self-contained

full rationale

The paper explicitly constructs ETGRS codes by appending three columns to a TGRS generator matrix, then derives necessary and sufficient conditions for MDS/AMDS via direct rank and distance analysis of the resulting generator matrix and its dual. NMDS conditions follow from the dual AMDS analysis, and non-GRS character is verified by the Schur product method with explicit examples. All steps use standard finite-field linear algebra on the stated parameters; no step reduces a claimed prediction or condition to a fitted input, self-definition, or load-bearing self-citation chain. The logical chain from construction to theorems is independent and externally verifiable.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard finite-field arithmetic and the definition of twisted GRS codes; no new free parameters or invented entities are indicated in the abstract.

axioms (1)
  • standard math Finite fields admit the evaluation and interpolation properties used for Reed-Solomon-type codes
    Invoked implicitly when constructing generator matrices and distance bounds

pith-pipeline@v0.9.0 · 5661 in / 1132 out tokens · 20902 ms · 2026-05-25T03:07:04.268742+00:00 · methodology

discussion (0)

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Reference graph

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