Near-factorizations of dihedral groups

Donald L. Kreher; Douglas R. Stinson; Maura B. Paterson

REVIEW 4 major objections 3 minor 1 cited by

Near-factorizations of dihedral groups

T0 review · 4 major / 3 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper proves that two published families of near-factorizations of dihedral groups are equivalent, and it constructs new nonequivalent examples in other nonabelian groups.

desk verdict Solid algebraic core with a real correction to the literature; computational sections need code or data before the uniqueness claims can be trusted. read the letter →

arxiv 2411.15884 v2 pith:J7FCGN2S submitted 2024-11-24 math.GR math.CO

Donald L. Kreher , Maura B. Paterson , Douglas R. Stinson This is my paper · ORCID

classification math.GRmath.CO MSC 20D6005B30

keywords near-factorizationdihedralgroupstrongexternaldifferencefamilygeneralizedequivalencePéchertransformcomputerenumerationnonequivalentconstructions

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Near-factorizations are pairs of subsets whose products cover every non-identity element of a finite group exactly once. This paper shows that two well-known infinite families of near-factorizations of dihedral groups, one from 1990 and one from 2008, are in fact equivalent under a single automorphism, contrary to a claim in the later paper. It also proves that $D_{(a^2+1)/2}$ admits nonequivalent near-factorizations for every composite odd $a$, and it uses computer search to exhibit nonequivalent examples in $D_{95}$, $D_5 \times \mathbb{Z}_5$, and a semidirect product $C_5^2 \rtimes C_2$. These results matter because genuine nonequivalence is the exception rather than the rule, and near-factorizations are equivalent to strong external difference families used in coding theory and cryptography.

What carries the argument

The central objects are near-factorizations $(A,B)$ of a group $G$ and the equivalence relation generated by $(A,B)\mapsto (f(A)h,\,h^{-1}f(B))$ for automorphisms $f$ and elements $h$. The paper's main identity is the explicit automorphism $f_{-d_1,0}$ of $D_n$, defined by $f(b)=b^{-d_1}$ and $f(a)=a$, which maps one published construction onto the other; the numerical congruence $\alpha d_1\equiv d_2\pmod n$ makes the image match exactly. The Pécher transform, which sends a symmetric near-factorization of $\mathbb{Z}_{2n}$ with odd $n$ through $\mathbb{Z}_2\times \mathbb{Z}_n$ to a strongly symmetric near-factorization of $D_n$, is analysed in both directions. The computer enumeration adapts a technique cited from the literature to dihedral groups and computes canonical forms under the equivalence action.

What would settle it

Independently recompute, with different software, the full list of canonical forms for all divisors $k$ of $2n-1$ in $D_n$ for $n=32$ and $n=95$; if any divisor yields more than one equivalence class, Theorem 2.10 is false. Also check the two reported $(9,21)$-near-factorizations of $D_{95}$ directly: if they land in the same equivalence class, the Section 4.2 claim of nonequivalence fails.

Watch

Extended reading notes

Core claim

The central claim is that two published constructions of $(2^k-1,2^k+1)$-near-factorizations of $D_{2^{2k-1}}$ are equivalent, contrary to an assertion in the 2008 paper that they were not. The equivalence is explicit: the automorphism $f_{-d_1,0}$, with $d_1=2^k+3$, carries the 1990 construction to the 2008 construction. As a consequence, any graph-based invariant such as the alternating property must agree on the two constructions, so the earlier claim that they differ is refuted. The paper further establishes that $D_{(a^2+1)/2}$ admits nonequivalent near-factorizations for every composite odd $a$, and it reports computer-assisted classifications showing uniqueness up to equivalence for $n\le 32$ and new nonequivalent pairs in $D_{95}$, $D_5 \times \mathbb{Z}_5$, and $C_5^2 \rtimes C_2$. For the cyclic-to-dihedral transform of Section 3, the paper proves that equivalence is preserved in the forward direction and, under explicit gcd conditions, also in the inverse direction.

Load-bearing premise

The paper's computer-assisted claims—the uniqueness classification for $n\le 32$ and the nonequivalence of the constructed pairs in $D_{95}$, $D_5 \times \mathbb{Z}_5$, and $C_5^2 \rtimes C_2$—rest on enumerations and equivalence tests that are described only by citing a technique, without code or output data, and the claimed automorphism group of $C_5^2 \rtimes C_2$ is asserted without proof.

Editorial extensions

If this is right

The 1990 and 2008 infinite families of $(2^k-1,2^k+1)$-near-factorizations of $D_{2^{2k-1}}$ are the same up to equivalence, so every invariant of near-factorizations, including the alternating-graph property, takes the same value on both.
For every $n\le 32$, each divisor $k$ of $2n-1$ supports exactly one equivalence class of $(k,(2n-1)/k)$-near-factorizations of $D_n$.
For every composite odd $a$, the dihedral group $D_{(a^2+1)/2}$ admits nonequivalent $(a,a)$-near-factorizations, giving an infinite family where nonequivalence actually occurs.
The Pécher transform maps equivalent symmetric near-factorizations of $\mathbb{Z}_{2n}$ to equivalent strongly symmetric near-factorizations of $D_n$, and under the stated gcd conditions the inverse map also preserves equivalence.
There exist nonequivalent $(7,7)$-near-factorizations in $D_5 \times \mathbb{Z}_5$ and in $C_5^2 \rtimes C_2$, and nonequivalent $(9,21)$-near-factorizations in $D_{95}$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because near-factorizations correspond precisely to two-set generalized strong external difference families, the equivalence and nonequivalence results transfer directly to GSEDFs, meaning that any design-theoretic use of the two families from 1990 and 2008 must treat them as the same object.
The $D_{95}$ example suggests that the gcd conditions in the inverse Pécher theorem are sufficient but not necessary; testing other parameter pairs would clarify whether the inverse transform preserves nonequivalence more generally.
The infinite family in Theorem 4.1 is built from blowup sequences for cyclic groups, so adapting the same blowup idea directly to nonabelian groups might yield further nonequivalent near-factorizations outside the dihedral family.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here's the short version: this paper earns its keep. It shows that two published construction families of near-factorizations in dihedral groups—de Caen et al. and Basco et al.—produce equivalent near-factorizations (Theorem 2.4), contradicting a claim in [1], and it gives the first infinite family of nonequivalent near-factorizations in dihedral groups (Theorem 4.1). The algebraic core is solid: the equivalence is explicit (via f_{-d1,0}) and checkable, and the Pecher-transform analysis is careful, with a preservation theorem and a partial converse under gcd conditions.

What is new is genuine. Beyond the correction, the new examples in D_41, D_95, D_5×Z_5, and C_5^2⋊C_2 are concrete, and the transform theorems are reusable. The main proofs are done from definitions, not fitted to conclusions.

The soft spots are real but mostly addressable. The computational claims—Theorem 2.10 (uniqueness for n ≤ 32), the two (9,21) classes in Z_190, the two (7,7) classes in D_5×Z_5 and in C_5^2⋊C_2—are not independently checkable from the preprint. The algorithm is only described as 'the technique from [9], adapted,' with no code, no output data, and no statement of how canonical forms or automorphism groups were computed. Section 4.3 has a more specific problem: the proof of aut(D_5×Z_5) appears to swap the roles of a and b, saying f(b) is an order-2 reflection when b has order 5. As written that proof is wrong; the stated automorphism group may still be correct, but the argument needs repair. Theorem 4.1's arithmetic-progression argument is informal, and the paper leans on the authors' own unpublished preprints [9] and [12] for enumeration and blowup machinery.

None of this undercuts the central structural theorems. Theorem 2.4 is a direct computation; Theorems 3.7 and 3.11 are carefully argued. I would send this to a serious referee. The computational sections should be treated as claims to verify, and the authors should be asked to release code or at least a detailed algorithm description. I'd cite this paper in my own work, and I'd want the Section 4.3 proof fixed before publication.

Referee Report

4 major / 3 minor

Summary. The paper studies near-factorizations of nonabelian groups, with emphasis on dihedral groups. After setting up equivalence of near-factorizations and reviewing the connection to generalized strong external difference families, the paper proves (Theorem 2.4) that the (2^k-1, 2^k+1)-near-factorizations of D_{2^{2k-1}} constructed by de Caen et al. and by Bascó et al. are equivalent via the explicit automorphism f_{-d_1,0}, contradicting a separation claimed in [1]. Section 3 analyzes the Pecher transform from symmetric near-factorizations of Z_{2n} to strongly symmetric near-factorizations of D_n for odd n: Theorem 3.7 shows equivalence is preserved, and Theorem 3.11 proves a partial converse under gcd conditions, yielding Corollary 3.12 for the (k,k) case. Section 4 constructs nonequivalent near-factorizations: an infinite family for D_{(a^2+1)/2} with composite odd a (Theorem 4.1), a computer-generated pair in D_95, and computer classifications showing exactly two equivalence classes in D_5×Z_5 and in C_5^2⋊_2 C_2.

Significance. The paper makes a useful contribution to a sparse literature. If the computational and infinite-family claims are fully supported, it supplies a nontrivial collection of nonequivalent near-factorizations in nonabelian groups and corrects the literature on the two dihedral constructions. I verified the algebra of Theorem 2.4; the explicit automorphism is a clean and checkable result. The Pecher-transform section is carefully developed and the partial converse in Theorem 3.11 is a genuine tool, despite one local proof error noted below. The main weaknesses are reproducibility: exact enumeration claims in Theorem 2.10 and Sections 4.2-4.4 are not backed by code, data, or a full algorithm, and the proof of Theorem 4.1 rests on an informal arithmetic-progression argument.

major comments (4)

[Section 4.1 (Theorem 4.1)] The nonequivalence argument for the blowup construction is not carried out. The proof asserts that any SEDF equivalent to (A_1,B_1) also contains arithmetic progressions of length a 'provided that we let the arithmetic sequence wrap around', and that the (j,j,k,k) blowup SEDFs 'do not contain arithmetic sequences of length a'. No definition of a wrap-around arithmetic progression of length a is given, and no proof is provided for the (j,j,k,k) case. Because this is the sole basis for the infinite family in Theorem 4.1, the authors should make the property precise (e.g., a subset of Z_{2n} of the form {x+tr:0≤t<a} with gcd(r,2n)=1) and prove both the invariance under the cyclic equivalence action and the failure for at least one of A_2,B_2 for all j,k>1.
[Theorem 2.10 and Sections 4.2-4.4] The exact computational claims are not reproducible. Theorem 2.10 asserts uniqueness up to equivalence for every n≤32; Section 4.2 asserts exactly two (9,21) classes in Z_190 (and hence, after the Pecher transform and a separate computer test, two classes in D_95); Section 4.3 asserts exactly two (7,7) classes in D_5×Z_5; Section 4.4 asserts exactly two classes in C_5^2⋊_2 C_2. In every case the only algorithmic description is that the technique of [9] was 'adapted to the setting of dihedral groups' (Section 2.1), and [9] is an unpublished preprint. No code, output data, search bounds, or descriptions of canonical forms and equivalence tests are provided. These claims are load-bearing: if the search missed an equivalence, the claimed classifications would fail. The paper should either include the programs and data or give a sufficiently detailed algorithm, including how aut(G) is computed for each group.
[Section 4.3] The proof that aut(D_5×C_5) ≅ aut(D_5)×aut(C_5) swaps the roles of f(a) and f(b). In the presentation a has order 2 and b has order 5, so f(a) must have order 2 and f(b) order 5; the text instead sets y=f(b)=ab^j (order 2) and x=f(a)=b^i c^k (order 5). The computation 'xyxy=e' that follows does not restore the correct order assignments. Although the stated automorphism group may be correct, this derivation is invalid and should be replaced, e.g., by starting with x=ab^j and y=b^i c^t and using (xy)^2=e to force t=0. Since the equivalence enumeration in Section 4.3 depends on this automorphism group, the correction is necessary.
[Section 3.11 (proof of Theorem 3.11, h=a case)] In the h=a case the map f(e,h)=(e+1,ih) is claimed to be an automorphism of Z_2×Z_n, but it is not: f(0,0)=(1,0)≠(0,0). The map is the affine map φ(e,h)=(e,ih) followed by translation by (1,0). Because (1,0) has order 2, the conclusion can be repaired by taking the automorphism φ and translation h=(1,0): one gets C^*=φ(A^*)+h and D^*=-h+φ(B^*). The current text should be corrected accordingly, since this proof underpins Corollary 3.12 and Section 4.1.

minor comments (3)

[Theorem 2.4] The displayed statement says 'D_{2^k-1}' but the proof and surrounding definitions use n=2^{2k-1}; the exponent should be corrected to D_{2^{2k-1}}.
[Definition 2.7 and references] The name 'Bocsó' appears in Definition 2.7, while the reference list uses 'Bascó'; please unify the spelling.
[References [9], [12]] The enumeration and blowup-sequence machinery are imported from unpublished preprints [9] and [12]. Please include arXiv identifiers and explicitly state which facts are assumptions from those papers, so a reader can check the dependencies.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: explicit automorphism and Pécher-transform proofs carry the main claims; self-cited preprints and unverified searches are reproducibility gaps, not circularity.

full rationale

After walking the derivation chain, I find no step in which a claimed prediction or theorem is identical to its input by construction. Theorem 2.4 is an explicit calculation: the paper defines A,B from [4] and A',B' from [1], sets f=f_{-d1,0}, proves f(A)=A' and f(B)=B' using modular identities (alpha*d1≡d2, s≡a0*d1, s≡(b0+1)*d2), and concludes equivalence. This is a direct verification, not an equivalence assumed in the definitions. The Pécher-transform results (Theorems 3.2, 3.7, 3.8, 3.11) are proved by explicit case analysis with the stated gcd hypotheses; Lemma 3.3 and Lemma 3.10 are proved from symmetric-set sum arguments. Section 4.1 gives explicit SEDFs and the nonequivalence argument (unions of arithmetic progressions) does not depend on a uniqueness theorem; the cited blowup sequences from [9,12] are explicit constructions rather than imported conclusions. The computational completeness claims (Theorem 2.10; Sections 4.2-4.4) and the automorphism-group descriptions are not supplied with code or full proofs, and Section 4.3's proof appears to swap images of a and b; however, these are reproducibility and correctness gaps, not circularity. There is no fitted parameter renamed as a prediction and no conclusion is used as a premise. Score 2 reflects the reliance on the authors' own unpublished preprints [9,12] for enumeration and blowup methods and the absence of independent computational artifacts; it does not reflect circular reasoning.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central theorems are proved from the paper's own definitions. The assembled axioms are prior results, some from unpublished preprints by the same authors, plus two computational or automorphism assumptions. No parameters are fitted; n, k, and l are construction parameters.

assumptions (4)

standard math Near-factorizations in abelian groups admit symmetric translates (Theorem 1.4 from [4]).
Used in Sections 4.1 and 4.2 to convert near-factorizations of Z_{2n} into symmetric ones before applying the Pecher transform.
domain assumption The blowup sequence method from [12] constructs valid SEDFs and near-factorizations.
Used in Theorem 4.1 to build two SEDFs in Z_{a^2+1} from sequences (a,a) and (j,j,k,k); [12] is a preprint by two of the authors.
domain assumption The enumeration and equivalence-testing technique from [9] is correct and complete.
The paper states that it used the technique from [9] to enumerate all nonequivalent near-factorizations of D_n for n <= 32 and to test nonequivalence in D_95 and D_5 x Z_5; no code or details are included.
domain assumption The listed automorphism generators for G = C_5^2 semidirect C_2 generate the full automorphism group.
Section 4.4 states generators x,y,z,w for aut(G) with no completeness proof; the equivalence classification depends on this.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Near-factorizations of dihedral groups." pith.science (2026). https://pith.science/paper/J7FCGN2S

@misc{pith2026241115884,
  author       = {Pith},
  title        = {Pith review of: Near-factorizations of dihedral groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/J7FCGN2S}},
  note         = {Machine review of arXiv:2411.15884}
}

read the original abstract

We investigate near-factorizations of nonabelian groups, concentrating on dihedral groups. We show that some known constructions of near-factorizations in dihedral groups yield equivalent near-factorizations. In fact, there are very few known examples of nonequivalent near-factorizations in dihedral or other nonabelian groups; we provide some new examples with the aid of the computer. We also analyse a construction for near-factorizations in dihedral groups from near-factorizations in cyclic groups, due to P\^{e}cher, and we investigate when nonequivalent near-factorizations can be obtained by this method.

Figures

Figures reproduced from arXiv: 2411.15884 by the authors.

**Figure 3.1.** The Pˆecher transform Example 3.1. A = {0, 1, 9} and B = {2, 5, 8} form a symmetric near-factorization of Z10. Applying φ, the associated (symmetric) near-factorization of Z2 × Z5 is (A ∗ , B∗ ) = ({(0, 0),(1, 1),(1, 4)}, {(0, 2),(1, 0),(0, 3)}). Then we use ψ to obtain the strongly symmetric near-factorization (A ′ , B′ ) = e, ab, ab4 [PITH_FULL_IMAGE:figures/full_fig_p014_3_1.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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

Cyclotomic construction of $\lambda$-fold near-factorizations of cyclic groups
math.CO 2025-07 conditional novelty 6.0 of 10

For primes p = 4n^4 + 12n^2 + 1, four explicit pairs of unions of order-8 cyclotomic cosets in F_p are shown to be (p-1)/16-fold near-factorizations.

Reference graph

Works this paper leans on

14 extracted references · 13 canonical work pages · cited by 1 Pith paper

[1]

Bacs´ o, L.cH´ ethelyi and P

G. Bacs´ o, L.cH´ ethelyi and P. Sziklai, New near-factorizations of ﬁnite groups, Studia Sci. Math. Hungar. 45 (2008), 493–510

work page 2008
[9]

Kreher, M.B

D.L. Kreher, M.B. Paterson and D.R. Stinson. Strong exte rnal diﬀerence families and classiﬁcation of α-valuations. Preprint. https://arxiv.org/abs/2406.09075 25

arXiv
[12]

Paterson and D.R

M.B. Paterson and D.R. Stinson. Circular external diﬀer ence families, graceful la- bellings and cyclotomy. Discrete Math. 347 (2024), #114103, 15 pp

work page 2024
[2]

J. Bao, L. Ji, R. Wei and Y. Zhang. New existence and nonexi stence results for strong external diﬀerence families Discrete Mathematics 341 (2018), 1798–1805

work page 2018
[3]

de Bruijn

N.G. de Bruijn. On number systems. Nieuw Archief voor Wiskunde (3) IV, (1956), 15–17

work page 1956
[4]

de Caen, D.A

D. de Caen, D.A. Gregory, I.G. Hughes and D.L. Kreher. Nea r-factors of ﬁnite groups. Ars Combin. 29 (1990), 53–63

work page 1990
[5]

Huczynska, C

S. Huczynska, C. Jeﬀerson and S. Nepˇ sinsk´ a. Strong exte rnal diﬀerence families in abelian and non-abelian groups. Cryptography and Communications 13 (2021), 331– 341

work page 2021
[6]

Huczynska and M.B

S. Huczynska and M.B. Paterson. Existence and non-exist ence results for strong exter- nal diﬀerence families. Discrete Mathematics 341 (2018), 87–95

work page 2018

Show all 14 references

[7]

Jedwab and S

J. Jedwab and S. Li. Construction and nonexistence of str ong external diﬀerence fam- ilies. J. Algebr. Comb. 49 (2019), 21–48

2019
[8]

Kreher, W.J

D.L. Kreher, W.J. Martin and D.R. Stinson. Uniqueness an d explicit computation of mates in near-factorizations. Preprint
[10]

X. Lu, X. Niu, and H. Cao. Some results on generalized str ong external diﬀerence families. Des. Codes Cryptogr. 86 (2018), 2857–2868

2018
[11]

Paterson and D.R

M.B. Paterson and D.R. Stinson. Combinatorial charact erizations of algebraic manip- ulation detection codes involving generalized diﬀerence fa milies. Discrete Math. 339 (2016), 2891–2906

2016
[13]

Pˆ echer

A. Pˆ echer. Cayley partitionable graphs and near-fact orizations of ﬁnite groups. Discr. Math. 276 (2004) 295–311

2004
[14]

J. Wen, M. Yang, F. Fu, and K. Feng. Cyclotomic construct ion of strong external diﬀerence families in ﬁnite ﬁelds. Des. Codes Cryptogr. 86 (2018), 1149–1159. 26

2018

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Bacs´ o, L.cH´ ethelyi and P

G. Bacs´ o, L.cH´ ethelyi and P. Sziklai, New near-factorizations of ﬁnite groups, Studia Sci. Math. Hungar. 45 (2008), 493–510

work page 2008

[2] [9]

Kreher, M.B

D.L. Kreher, M.B. Paterson and D.R. Stinson. Strong exte rnal diﬀerence families and classiﬁcation of α-valuations. Preprint. https://arxiv.org/abs/2406.09075 25

arXiv

[3] [12]

Paterson and D.R

M.B. Paterson and D.R. Stinson. Circular external diﬀer ence families, graceful la- bellings and cyclotomy. Discrete Math. 347 (2024), #114103, 15 pp

work page 2024

[4] [2]

J. Bao, L. Ji, R. Wei and Y. Zhang. New existence and nonexi stence results for strong external diﬀerence families Discrete Mathematics 341 (2018), 1798–1805

work page 2018

[5] [3]

de Bruijn

N.G. de Bruijn. On number systems. Nieuw Archief voor Wiskunde (3) IV, (1956), 15–17

work page 1956

[6] [4]

de Caen, D.A

D. de Caen, D.A. Gregory, I.G. Hughes and D.L. Kreher. Nea r-factors of ﬁnite groups. Ars Combin. 29 (1990), 53–63

work page 1990

[7] [5]

Huczynska, C

S. Huczynska, C. Jeﬀerson and S. Nepˇ sinsk´ a. Strong exte rnal diﬀerence families in abelian and non-abelian groups. Cryptography and Communications 13 (2021), 331– 341

work page 2021

[8] [6]

Huczynska and M.B

S. Huczynska and M.B. Paterson. Existence and non-exist ence results for strong exter- nal diﬀerence families. Discrete Mathematics 341 (2018), 87–95

work page 2018