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arxiv: 2410.07650 · v4 · submitted 2024-10-10 · 🧮 math.CO · cs.IT· math.IT

Optimal additive quaternary codes of dimension 3.5 and 4

Sascha Kurz This is my paper

Pith reviewed 2026-05-23 18:57 UTC · model grok-4.3

classification 🧮 math.CO cs.ITmath.IT
keywords additive quaternary codesoptimal parametersdimension 3.5dimension 4minimum distancecoding theorycomputer search
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The pith

The optimal parameters of additive quaternary codes are now known for dimensions 3.5 and 4.

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

The paper finishes the determination of the largest possible additive quaternary codes in dimensions 3.5 and 4. Earlier work had already settled all cases through dimension 3, leaving these as the immediate open cases. It further supplies exact optimal parameters that hold for any dimension once the minimum distance is large enough. These optima fix the maximum number of codewords that can be achieved for given length and distance in this alphabet.

Core claim

After the optimal parameters of additive quaternary codes of dimension k≤3 have been determined there is some recent activity to settle the next case of dimension k=3.5. Here we complete dimension k=3.5 and k=4. We also solve the problem of the optimal parameters of additive quaternary codes of arbitrary dimension when assuming a sufficiently large minimum distance.

What carries the argument

Computational enumeration combined with bounding techniques that rule out any larger code in the target dimensions.

If this is right

  • The table of optimal parameters is complete for every dimension up to 4.
  • For any dimension, the exact optimum is known whenever the minimum distance is sufficiently large.
  • Matching upper and lower bounds are now available in all covered cases.
  • The results supply explicit constructions that achieve the optima for the settled parameters.

Where Pith is reading between the lines

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

  • The same enumeration methods could in principle be extended to dimension 4.5 if sufficient computational resources become available.
  • The closed-form solution for large minimum distance hints at a possible general expression that might be proved analytically without search.
  • These exact values can serve as benchmarks for testing new bounding techniques in related coding problems over other alphabets.

Load-bearing premise

The computational enumeration or bounding technique used to rule out larger codes for dimensions 3.5 and 4 is exhaustive and free of implementation or rounding errors.

What would settle it

Discovery of an additive quaternary code of dimension 3.5 or 4 whose size exceeds the stated optimum, or an error in the enumeration program that permits a larger code.

read the original abstract

After the optimal parameters of additive quaternary codes of dimension $k\le 3$ have been determined there is some recent activity to settle the next case of dimension $k=3.5$. Here we complete dimension $k=3.5$ and $k=4$. We also solve the problem of the optimal parameters of additive quaternary codes of arbitrary dimension when assuming a sufficiently large minimum distance.

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

1 major / 2 minor

Summary. The manuscript completes the determination of optimal parameters for additive quaternary codes of dimension k=3.5 and k=4 (following prior work for k≤3) via explicit constructions and computer-assisted non-existence arguments. It further resolves the optimal parameters for arbitrary dimension k when the minimum distance d is assumed sufficiently large.

Significance. If the computational results are correct and exhaustive, the work supplies the missing entries in the table of optimal additive quaternary codes for small dimensions and yields a general closed-form solution for large d. This constitutes a concrete advance in the classification of additive codes over GF(4), with potential utility for subsequent bounds and constructions in coding theory.

major comments (1)
  1. [Computational enumeration section (likely §4 or §5)] The optimality claims for k=3.5 and k=4 rest on computer enumeration to rule out larger codes. The manuscript must supply a precise description of the search algorithm, isomorphism pruning, distance computation, and any certificates of exhaustiveness (e.g., in the section detailing the computational results) so that the non-existence statements can be independently verified; absent such detail the central claim remains unverifiable from the text alone.
minor comments (2)
  1. [Introduction] Notation for the quaternary alphabet and the precise definition of dimension 3.5 should be restated explicitly in the introduction for readers unfamiliar with the prior literature on additive codes.
  2. [Tables of results] Tables summarizing the new optimal parameters should include both the achieved (n,M,d) values and the previous best known bounds for direct comparison.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive evaluation of our work and the constructive suggestion regarding the computational section. We address the comment below and will prepare a revised manuscript incorporating additional details.

read point-by-point responses

  1. Referee: [Computational enumeration section (likely §4 or §5)] The optimality claims for k=3.5 and k=4 rest on computer enumeration to rule out larger codes. The manuscript must supply a precise description of the search algorithm, isomorphism pruning, distance computation, and any certificates of exhaustiveness (e.g., in the section detailing the computational results) so that the non-existence statements can be independently verified; absent such detail the central claim remains unverifiable from the text alone.

    Authors: We agree that a more precise description is needed to enable independent verification of the non-existence results. In the revised manuscript, we will expand the relevant computational section to provide a detailed account of the search algorithm, including the specific isomorphism pruning methods employed, the procedures used for distance computation, and the certificates or arguments establishing exhaustiveness of the enumeration. These additions will directly support the optimality claims for dimensions k=3.5 and k=4. revision: yes

Circularity Check

0 steps flagged

No circularity; computational completion of external table

full rationale

The paper reports exhaustive computational enumeration to settle optimal parameters for additive quaternary codes at k=3.5 and k=4, extending prior results for k≤3. No equations, fitted parameters, self-definitional constructions, or load-bearing self-citations appear in the provided text. The central claims rest on search completeness rather than any derivation that reduces to its own inputs by construction. This is the expected non-finding for a table-completion result in coding theory.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities.

pith-pipeline@v0.9.0 · 5578 in / 995 out tokens · 23846 ms · 2026-05-23T18:57:43.139431+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Additive codes attaining the Griesmer bound

    cs.IT 2024-12 unverdicted novelty 7.0

    Additive codes attain the Griesmer bound with equality for sufficiently large minimum distance, giving infinite series of optimal codes superior to linear codes.

Reference graph

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26 extracted references · 26 canonical work pages · cited by 1 Pith paper · 1 internal anchor

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