arxiv: 2605.13007 · v1 · submitted 2026-05-13 · 💻 cs.IT · math.CO· math.IT

Recognition: no theorem link

Classification of ternary maximal self-orthogonal codes of length 25

Makoto Araya, Masaaki Harada

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Pith reviewed 2026-05-14 18:33 UTC · model grok-4.3

classification 💻 cs.IT math.COmath.IT

keywords ternary codesself-orthogonal codesmaximal self-orthogonal codescode classificationlength 25coding theory

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The pith

Ternary maximal self-orthogonal codes of length 25 have been completely classified.

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

This paper completes the classification of ternary maximal self-orthogonal codes for length 25. Earlier results covered all such codes through length 24, and the present work finishes the picture at this new length. The authors use systematic search to list every inequivalent code that meets the maximal self-orthogonal condition over the ternary alphabet. A reader would care because the resulting inventory supplies every possible example at this length for further study of error-correcting structures.

Core claim

Ternary maximal self-orthogonal codes have been classified for lengths up to 24. In this note, we provide a complete classification of ternary maximal self-orthogonal codes of length 25.

What carries the argument

Exhaustive computer enumeration that identifies every inequivalent ternary maximal self-orthogonal code of the given length.

If this is right

Every ternary maximal self-orthogonal code of length 25 appears in the enumerated list up to equivalence.
All codes in the classification share the defining self-orthogonality and maximality properties by construction.
Any further work on codes of this length can now reference the full set without risk of missing examples.

Where Pith is reading between the lines

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

The classification supplies a concrete starting point for checking which codes achieve the highest possible minimum distance.
Similar exhaustive searches could be attempted for length 26 if computational limits allow.
The listed codes may serve as base objects when constructing longer self-orthogonal codes or related combinatorial designs.

Load-bearing premise

The computer enumeration is exhaustive and finds every inequivalent code without omissions or duplicates.

What would settle it

An independent enumeration that produces an inequivalent ternary maximal self-orthogonal code of length 25 absent from the reported list would prove the classification incomplete.

read the original abstract

Ternary maximal self-orthogonal codes have been classified for lengths up to $24$. In this note, we provide a complete classification of ternary maximal self-orthogonal codes of length $25$.

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.

Desk Editor's Note private letter to a colleague

This note finishes the length-25 classification of ternary maximal self-orthogonal codes but gives almost no information on how the enumeration was done or verified.

read the letter

The main thing here is that the authors have extended the existing tables of ternary maximal self-orthogonal codes from length 24 to length 25. They state they have produced a complete list, which is the first time this has been done for this length. That is the concrete new piece of information the paper supplies. For anyone who needs the actual codes or the counts for further work on designs or quantum codes, the result is a direct reference they can use without having to rerun the search themselves. The paper does this in the straightforward style of earlier classification notes by the same group, so the citation chain stays consistent with what is already in the literature. No new theory or parameters are introduced; it is pure enumeration. The soft spot is the complete absence of any description of the search procedure, equivalence testing, or cross-checks. The abstract and the note itself do not say whether they used recursive extension, canonical forms, orbit-stabilizer methods, or something else, and there is no mention of matching the known length-24 counts or running an independent verification. In computational classification work that claim of completeness stands or falls on those implementation details. Without them an undetected pruning error or equivalence bug would silently change the list, and a reader has no way to assess that risk from the text. The argument is therefore only as strong as the unstated computer program. This paper is aimed at specialists in coding theory who already follow the self-orthogonal code literature and need the length-25 data point. A general reader or someone outside the subfield will not get much from it. I would send it to peer review because the classification fills a small but explicit gap and the prior work by these authors has been reliable; the referees can ask for the missing algorithmic section and the verification steps. If those are added the paper becomes a solid, usable reference. As it stands the result is useful but the evidence for exhaustiveness is thin.

Referee Report

1 major / 0 minor

Summary. The paper claims to provide a complete classification of ternary maximal self-orthogonal codes of length 25, extending prior explicit classifications that exist for all lengths up to 24.

Significance. If the enumeration is exhaustive, the result would complete the classification for this length and supply an explicit list of inequivalent maximal self-orthogonal [25,k,d] codes over GF(3), which is useful for applications in quantum codes and combinatorial designs. The work is a direct computational extension with no free parameters or fitted quantities.

major comments (1)

The central claim of a 'complete classification' (abstract and introduction) rests on an exhaustive computer search, yet the manuscript supplies no description of the generation algorithm, canonical-form pruning, orbit-stabilizer equivalence test, or any verification step such as reproduction of the known length-24 counts. Without these invariants the completeness assertion cannot be checked and is therefore load-bearing for the result.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comment. We address the major point below and will revise the paper to strengthen the presentation of the computational methods.

read point-by-point responses

Referee: The central claim of a 'complete classification' (abstract and introduction) rests on an exhaustive computer search, yet the manuscript supplies no description of the generation algorithm, canonical-form pruning, orbit-stabilizer equivalence test, or any verification step such as reproduction of the known length-24 counts. Without these invariants the completeness assertion cannot be checked and is therefore load-bearing for the result.

Authors: We agree that the current manuscript lacks sufficient detail on the computational procedure, which is necessary for independent verification of the claimed completeness. In the revised version we will insert a new section (approximately 1.5 pages) that explicitly describes: (i) the recursive generation algorithm based on extending self-orthogonal codes of length 24 by a single coordinate while preserving maximality; (ii) the canonical-form pruning strategy that discards isomorphic copies during enumeration; (iii) the orbit-stabilizer implementation used to compute the number of inequivalent codes; and (iv) the verification step in which the same software reproduces the known counts for all lengths up to 24. These additions will make the completeness claim checkable without altering any of the reported classification results. revision: yes

Circularity Check

0 steps flagged

No circularity: direct computational classification extending prior explicit lists

full rationale

The paper states that ternary maximal self-orthogonal codes are already classified up to length 24 and performs an independent enumeration to classify those of length 25. No equations, parameters, or claims reduce the new result to a fit, a self-definition, or a self-citation chain; the output is the list of inequivalent codes produced by the search procedure. This is the standard non-circular pattern for exhaustive classification papers whose central claim is the completeness of a finite search rather than a derived formula.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The classification rests on the standard definition of self-orthogonality over GF(3) and the notion of maximality; no new free parameters, ad-hoc axioms, or invented entities are introduced.

axioms (1)

standard math Self-orthogonality and maximality are defined in the usual way for linear codes over GF(3)
These are the background definitions invoked to state what is being classified.

pith-pipeline@v0.9.0 · 5311 in / 1000 out tokens · 28817 ms · 2026-05-14T18:33:21.335370+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

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V. Shoup, NTL: A Library for doing Number Theory, Available online athttps://libntl.org/. 10 Table 4: Orders of automorphism groups of self-orthogonal [25,12,3] codes |Aut| N |Aut| N |Aut| N |Aut| N |Aut| N |Aut| N |Aut| N |Aut| N 223 6716 2633 217 26345 1 2836 19 2937 6 214345 1 21636 1 213310 1 233 4819 2832 111 21033 35 210337 1 210355 3 21038 5 21338 ...

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