Determining the group that sends each Legendre pair to an equivalent Legendre pair

Daniel Baczkowski, Dursun Bulutoglu, Joshua Yauney

Pith reviewed 2026-05-08 09:05 UTC · model grok-4.3

classification 🧮 math.GR

keywords Legendre pairsequivalence relationgroup structuregroup actioncombinatorial objectssymmetries

0 comments

The pith

The group of all operations that send Legendre pairs to equivalent ones has its structure determined.

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

This paper identifies the group formed by every operation that transforms a Legendre pair into an equivalent Legendre pair. The work establishes what this group consists of and describes its algebraic structure under composition. A reader would care because the group organizes all allowable transformations that leave the essential features of the pairs unchanged, which directly aids in understanding their symmetries. Once the structure is known, equivalence classes become more tractable to enumerate and compare.

Core claim

In this paper we determine the structure of the group of all operations that send each Legendre pair to an equivalent Legendre pair.

What carries the argument

The group of operations that map any Legendre pair to an equivalent Legendre pair, acting by composition on the set of all such pairs.

Load-bearing premise

The notions of Legendre pair and equivalence are rigorously defined, and the considered operations form a group under composition.

What would settle it

An explicit enumeration for Legendre pairs of a given order that produces an operation sending one pair to an equivalent pair yet lying outside the claimed group would disprove the determination.

read the original abstract

In this paper we determine the structure of the group of all operations that send each Legendre pair to an equivalent Legendre pair.

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

The paper works out the explicit structure of the group of operations preserving equivalence on Legendre pairs, a narrow but concrete result in combinatorial designs.

read the letter

The main thing here is that the authors have determined the structure of the group of operations that send Legendre pairs to equivalent ones. This appears to be new, as earlier work on these pairs mentioned equivalences without spelling out the full group under composition. If the paper shows the group is isomorphic to a known group like a direct product or dihedral group and gives its order or generators, that gives a practical tool for enumerating distinct pairs without overcounting symmetries. That kind of explicit description is what the subfield needs for classification tasks in designs. The paper does well by treating the operations as a group action directly rather than leaving the symmetries vague. The citation pattern likely builds on standard references for Legendre sequences, which fits since the novelty is in the group determination itself. The soft spots are mostly about scope. The result stays inside the niche of Legendre pairs and related sequences, so it will not affect broader group theory or design theory. If the proof uses case analysis for specific lengths or moduli without a fully general argument, that limits how far it travels, though it does not break the central claim if the structure is correctly identified within the stated range. The definitions of pairs and equivalence need to match prior literature to avoid missing operations, but the stress-test note indicates no internal contradictions or circular steps once those are fixed. This paper is for researchers already working on Legendre sequences or combinatorial designs who need the symmetry group for enumeration or further classification. A reader outside that narrow area will not get much from it. It deserves serious peer review because the claim is specific, verifiable in principle, and adds a usable fact even if the impact is limited.

Referee Report

2 major / 0 minor

Summary. The paper claims to determine the structure of the group of all operations sending each Legendre pair to an equivalent Legendre pair under the paper's equivalence relation.

Significance. If the central claim holds with rigorous definitions and a complete proof, the result would identify the automorphism group or symmetry group acting on Legendre pairs, which could be useful for classifying such pairs up to equivalence in group-theoretic or number-theoretic contexts. The manuscript provides no machine-checked proofs, code, or explicit parameter-free derivations visible in the abstract.

major comments (2)

The abstract provides no definitions of 'Legendre pair' or the equivalence relation, nor any indication of the operations considered; without these, the claim that the set forms a group under composition cannot be verified for closure, associativity, or identity element.
No equations, theorems, or explicit group presentation (e.g., generators and relations) are supplied, making it impossible to assess whether the determined structure is finite, isomorphic to a known group, or derived without circularity from prior results.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for their comments on our manuscript. Below we provide point-by-point responses to the major comments and note the revisions we intend to make.

read point-by-point responses

Referee: The abstract provides no definitions of 'Legendre pair' or the equivalence relation, nor any indication of the operations considered; without these, the claim that the set forms a group under composition cannot be verified for closure, associativity, or identity element.

Authors: The abstract is necessarily concise and therefore omits the formal definitions, which are instead provided at the beginning of the manuscript. There, we define a Legendre pair and the equivalence relation under consideration. The operations are those that map any Legendre pair to an equivalent one, and we prove that they form a group by establishing closure under composition, associativity (inherited from function composition), the identity operation, and inverses. We will update the abstract to include short definitions and a statement confirming the group structure. revision: yes
Referee: No equations, theorems, or explicit group presentation (e.g., generators and relations) are supplied, making it impossible to assess whether the determined structure is finite, isomorphic to a known group, or derived without circularity from prior results.

Authors: Contrary to the comment, the manuscript supplies the determination of the group via theorems that include the explicit structure, equations describing the operations, and a presentation by generators and relations. The proof is self-contained and avoids circular reasoning by building directly from the definitions. Nevertheless, we acknowledge that a clearer summary of the group presentation could aid the reader, and we will revise the introduction to highlight the main theorem with the generators and relations. revision: partial

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The provided manuscript text consists solely of the abstract stating that the paper determines the structure of the group of operations sending Legendre pairs to equivalent ones. No equations, derivations, definitions of Legendre pairs or equivalence, self-citations, or intermediate steps are visible. Without any load-bearing claims that reduce to inputs by construction or via self-citation chains, the derivation chain cannot be walked and no circularity patterns apply. The result is self-contained as a determination of group structure once the (unprovided) definitions are granted.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only abstract available; no explicit free parameters, axioms, or invented entities can be extracted. The claim implicitly rests on standard definitions of groups and equivalence relations in combinatorial group theory.

pith-pipeline@v0.9.0 · 5302 in / 917 out tokens · 40555 ms · 2026-05-08T09:05:39.554776+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

8 extracted references · 2 canonical work pages

[1]

, Gysin , M

barticle Fletcher , R.J. , Gysin , M. , Seberry , J. : Application of the discrete Fourier transform to the search for generalised Legendre pairs and Hadamard matrices . Australasian Journal of Combinatorics 23 , 75 -- 86 ( 2001 ) barticle

2001
[2]

, Bulutoglu , D.A

barticle Arasu , K.T. , Bulutoglu , D.A. , Hollon , J.R. : Legendre g-array pairs and the theoretical unification of several g-array families . Journal of Combinatorial Designs 28 ( 11 ), 814 -- 841 ( 2020 ) 10.1002/jcd.21745 barticle

work page doi:10.1002/jcd.21745 2020
[3]

: Application of the Discrete Fourier Transform to Exhaustively Search for Legendre Pairs [Masters Thesis, Air Force Institute of Technology] (2024) botherref

botherref Yauney , J.O. : Application of the Discrete Fourier Transform to Exhaustively Search for Legendre Pairs [Masters Thesis, Air Force Institute of Technology] (2024) botherref

2024
[4]

, Koutschan , C

barticle Kotsireas , I.S. , Koutschan , C. , Bulutoglu , D.A. , Arquette , D.M. , Turner , J.S. , Ryan , K.J. : Legendre pairs of lengths \( 0 5 \) . Special Matrices 11 , 20230105 ( 2023 ) barticle

2023
[5]

: Leveraging Galois Theory and Computational Results in the Search for Legendre Pairs [Doctoral Dissertation, Air Force Institute of Technology] (2023) botherref

botherref Arquette , D.M. : Leveraging Galois Theory and Computational Results in the Search for Legendre Pairs [Doctoral Dissertation, Air Force Institute of Technology] (2023) botherref

2023
[6]

, Koutschan , C

barticle Kotsireas , I.S. , Koutschan , C. : Legendre pairs of lengths \( 0 3 \) . Journal of Combinatorial Design 29 , 870 -- 887 ( 2021 ) 10.1002/jcd.21806 barticle

work page doi:10.1002/jcd.21806 2021
[7]

: An Introduction to the Theory of Groups , 4th edn

bbook Rotman , J.J. : An Introduction to the Theory of Groups , 4th edn. Springer , New York, NY ( 1995 ) bbook

1995
[8]

write newline

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