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arxiv: 2604.11350 · v2 · submitted 2026-04-13 · 💻 cs.IT · math.IT

Generalized Roth--Lempel Codes: NMDS Characterization, Hermitian Self-Orthogonality, and Quantum Constructions

Qi Liu , Xuefei Wu , Yingchun Cheng , Haiyan Zhou This is my paper

Pith reviewed 2026-05-10 16:10 UTC · model grok-4.3

classification 💻 cs.IT math.IT
keywords Generalized Roth-Lempel codesNMDS codesHermitian self-orthogonalQuantum error-correcting codesSingleton boundMDS codesClassical codesQuantum constructions
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The pith

Generalized Roth-Lempel codes supply explicit NMDS conditions and Hermitian self-orthogonal families that produce quantum codes attaining the Singleton bound minus one.

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

The paper supplies necessary and sufficient conditions that identify when two main subclasses of generalized Roth-Lempel codes are near-MDS. It then produces four families of Hermitian self-orthogonal codes drawn from the GRL framework, two of them NMDS and carrying parameters outside earlier lists. From these self-orthogonal objects it derives four families of quantum codes, two of which are infinite families of quantum NMDS codes that meet the quantum Singleton bound minus one. A sympathetic reader would care because the constructions fill documented gaps between classical algebraic codes and quantum error correction while generating many new or improved quantum parameters.

Core claim

We give explicit necessary and sufficient conditions for the NMDS property of the two most widely used subclasses of GRL codes. We construct four new families of Hermitian self-orthogonal codes from GRL codes; two of these families are NMDS with parameters not covered by existing Hermitian self-orthogonal NMDS codes. Based on the proposed Hermitian self-orthogonal GRL codes, we construct four families of quantum GRL codes, including two infinite families of quantum NMDS codes that attain the quantum Singleton bound minus one.

What carries the argument

The generalized Roth-Lempel framework that unifies Roth-Lempel codes under a flexible algebraic structure, together with the explicit parameter choices that enforce Hermitian self-orthogonality while preserving distance and dimension.

Load-bearing premise

The specific parameter choices inside the GRL framework make the Hermitian self-orthogonality condition hold without lowering the claimed distance or dimension.

What would settle it

A concrete parameter tuple from one of the four constructed families for which the resulting quantum code has minimum distance strictly less than the distance stated in the paper.

read the original abstract

In their seminal 1989 work (IEEE Trans. Inf. Theory 35(3):655-657), Roth and Lempel constructed a well-known family of non-Reed-Solomon maximum distance separable (MDS) codes. For decades, this family of codes has attracted extensive research attention due to its algebraic structure, low-complexity decoding, and broad applications in cryptography and data storage. Most recently, in 2025, the generalized Roth-Lempel (GRL) framework unifies Roth-Lempel codes and its extensions under a flexible algebraic structure. However, explicit criteria for the near-MDS (NMDS) property of GRL codes have not been established, and no systematic construction of Hermitian self-orthogonal GRL codes has been reported, limiting their deployment in classical and quantum error correction. In this work, we make three contributions to address these gaps. First, we give explicit necessary and sufficient conditions for the NMDS property of the two most widely used subclasses of GRL codes. Second, we construct four new families of Hermitian self-orthogonal codes from GRL codes. Two of these families are NMDS, with parameters not covered by existing Hermitian self-orthogonal NMDS codes. Third, based on the proposed Hermitian self-orthogonal GRL codes, we construct four families of quantum GRL codes, including two infinite families of quantum NMDS codes that attain the quantum Singleton bound minus one. Compared to the known quantum error-correcting codes, we obtain many new or improved quantum error-correcting codes. This work bridges the gap between classical GRL code families and quantum error-correction applications.

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 provides explicit necessary and sufficient conditions for the near-MDS (NMDS) property of two subclasses of generalized Roth-Lempel (GRL) codes. It constructs four families of Hermitian self-orthogonal GRL codes (two of them NMDS with novel parameters). These are then used to derive four families of quantum GRL codes, including two infinite families of quantum NMDS codes that attain the quantum Singleton bound minus one, yielding new or improved quantum error-correcting codes.

Significance. If the algebraic conditions and direct verifications hold, the work is significant for bridging classical GRL codes with quantum error correction. The explicit NMDS criteria and the two infinite families of quantum NMDS codes meeting the quantum Singleton bound minus one expand the known parameter sets and provide systematic constructions not covered by prior Hermitian self-orthogonal NMDS codes.

minor comments (2)
  1. [Abstract] Abstract: the year '2025' for the introduction of the GRL framework should be confirmed against the actual preprint or publication date for accuracy.
  2. The manuscript would benefit from including at least one small explicit example (e.g., small q and n) that verifies both the NMDS condition and the Hermitian self-orthogonality for one of the constructed families, to aid reader verification.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, recognition of the significance of our explicit NMDS criteria for GRL codes, the new Hermitian self-orthogonal families, and the resulting quantum NMDS constructions attaining the quantum Singleton bound minus one. The recommendation for minor revision is noted. As the report lists no specific major comments, we have no individual points to address point-by-point and will prepare a revised manuscript incorporating any minor editorial suggestions from the editor.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper supplies explicit algebraic necessary-and-sufficient conditions for the NMDS property of two GRL subclasses, constructs four new Hermitian self-orthogonal families (two NMDS) by direct parameter restrictions that preserve distance and dimension, and derives the four quantum GRL families from those conditions. None of the load-bearing steps reduces by definition, by fitted-parameter renaming, or by a self-citation chain to the input data; the quantum Singleton-bound-minus-one attainment follows from the stated Hermitian inner-product verification and the GRL framework parameters. The derivation chain is therefore self-contained against external algebraic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the work rests on standard algebraic properties of finite fields and the Hermitian inner product; no new entities are introduced and no free parameters are fitted to data.

axioms (1)
  • standard math Finite-field arithmetic and the definition of the Hermitian inner product over extension fields
    Invoked implicitly when defining self-orthogonality and code parameters.

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Reference graph

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