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arxiv: 2606.08964 · v1 · pith:E46FOAI7new · submitted 2026-06-08 · 💻 cs.IT · math.IT

Embedding linear codes over Z4 into self-orthogonal codes

Junmin An , Jon-Lark Kim , San Ling This is my paper

Pith reviewed 2026-06-27 15:05 UTC · model grok-4.3

classification 💻 cs.IT math.IT
keywords self-orthogonal embeddingZ4-linear codesPreparata codesdoubly even codesbinary linear codesLee distancecode construction
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The pith

The exact length of the shortest self-orthogonal embedding is determined for quaternary Preparata codes and classified for all binary linear codes.

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

The paper examines the problem of embedding linear codes over the integers modulo 4 into self-orthogonal codes of minimal possible length. It establishes tight bounds on this length and finds the exact value under certain conditions that the codes satisfy. One such family is the quaternary Preparata codes, for which the minimal embedding length is computed precisely. To support this, the paper also provides a complete case-by-case classification of the shortest doubly even self-orthogonal embeddings for every binary linear code. When the embedding length of a free Z4 code equals that of its residue code, an algorithm generates all shortest embeddings, resulting in the discovery of twelve Z4 codes with better minimum Lee distances than previously known.

Core claim

Under the conditions specified, the shortest self-orthogonal embedding length over Z4 equals a value determined by the code's parameters, as shown exactly for the Preparata codes; moreover, the shortest doubly even self-orthogonal embedding length for any binary linear code is fully classified according to its properties in every case. When these lengths match for a free Z4 code and its residue code, all shortest embeddings can be constructed via an explicit algorithm.

What carries the argument

The self-orthogonal embedding of a code, an extension by added coordinates that renders the code self-orthogonal while minimizing total length, linked to the residue code over F2 for the doubly even case.

If this is right

  • The minimal embedding length for quaternary Preparata codes is known exactly.
  • The shortest doubly even self-orthogonal embedding length is classified for every binary linear code.
  • An algorithm constructs all shortest self-orthogonal embeddings for qualifying free Z4 codes.
  • Twelve new Z4-linear codes are obtained with minimum Lee distances higher than those in existing databases.

Where Pith is reading between the lines

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

  • The classification for binary codes supplies a template that could be tested on codes over other rings.
  • The algorithm can be run on additional free Z4 codes to search for further improvements in minimum distance.
  • Knowing the exact lengths allows direct verification that a candidate embedding achieves the minimum without checking longer candidates.

Load-bearing premise

The exact lengths are determined only when the code meets the specific conditions outlined for the bounds to become equalities.

What would settle it

A counterexample would be a quaternary Preparata code that admits a self-orthogonal embedding of length shorter than the exact value established in the paper.

read the original abstract

The purpose of this paper is to investigate the self-orthogonal embedding problem for linear codes over Z4. We propose several tight bounds on the length of the shortest self-orthogonal embedding over Z4, and determine the exact shortest self-orthogonal embedding length under specific conditions. As an example satisfying these conditions, we establish the exact length of the shortest self-orthogonal embedding for the quaternary Preparata codes. Furthermore, to establish these results, we completely classify the exact length of the shortest doubly even self-orthogonal embedding for binary linear codes in every possible case. Finally, when the shortest self-orthogonal embedding length of a given free code over Z4 is equal to the shortest doubly even self-orthogonal embedding length of its residue code, we present an algorithm to construct all possible shortest self-orthogonal embeddings. With our algorithm, we found twelve linear codes over Z4 whose minimum Lee distances are higher than those of the Z4-linear codes in Aydins database.

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 studies self-orthogonal embeddings of linear codes over Z4. It derives tight bounds on the minimal length of such embeddings, obtains exact lengths under explicitly stated conditions (including an exact determination for the quaternary Preparata codes), supplies a complete case-by-case classification of the shortest doubly-even self-orthogonal embedding lengths for every binary linear code, and gives a constructive algorithm that produces all shortest embeddings whenever the Z4 length equals the doubly-even length of the residue code. The algorithm is used to exhibit twelve Z4-linear codes whose minimum Lee distances exceed those in Aydin's database.

Significance. If the stated classifications and constructions are correct, the work resolves the shortest doubly-even self-orthogonal embedding problem for binary codes in full generality and supplies exact results for an important family of quaternary codes. The algorithmic construction and the twelve new codes constitute concrete, reproducible contributions to the coding-theory literature.

minor comments (2)
  1. The abstract states that the classification covers 'every possible case' for binary codes; the introduction or §2 should explicitly list the cases (e.g., by dimension, minimum distance, or residue-code type) so that a reader can verify completeness without reconstructing the partition from the proofs.
  2. The algorithm in the final section is presented at a high level; pseudocode or a worked example for one of the twelve new codes would make the construction reproducible from the text alone.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the thorough and positive summary of our work, the assessment of its significance, and the recommendation for minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The abstract and summary describe constructive bounds, an exhaustive classification of embedding lengths under explicit conditions, and an algorithm that produces new codes with improved distances. No equations, self-citations, or fitted parameters are shown reducing the claimed lengths or classifications to the inputs by definition. The results are presented as independent of prior author work in a load-bearing way, and the algorithm is explicitly constructive rather than tautological. This is the normal case of a self-contained coding-theory paper.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are identifiable from the given text.

pith-pipeline@v0.9.1-grok · 5693 in / 1190 out tokens · 26044 ms · 2026-06-27T15:05:44.727598+00:00 · methodology

discussion (0)

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Reference graph

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