Uniqueness and explicit computation of mates in near-factorizations

Donald L. Kreher; Douglas R. Stinson; William J. Martin

REVIEW 1 major objections 4 minor 1 cited by

Uniqueness and explicit computation of mates in near-factorizations

T0 review · 1 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read In a near-factorization of a finite group, the mate of a set is unique and can be computed directly from a matrix formula.

desk verdict Clean, honest paper: the unique-mate formula and the sparse-system algorithm are real contributions; the sub-200 nonexistence claim rests on computer searches that need documentation before the full claim is checkable. read the letter →

arxiv 2411.15890 v1 pith:2NZEWTJQ submitted 2024-11-24 math.GR math.CO

Donald L. Kreher , William J. Martin , Douglas R. Stinson This is my paper · ORCID

classification math.GRmath.CO MSC 05B10

keywords near-factorizationmateuniquenessnoncyclicabeliangroupsincidencematrixstrongcircularexternaldifferencefamiliesexhaustivecomputersearchindexlambda

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

The paper establishes that, in a near-factorization of a finite group, a given subset can have at most one 'mate,' and that the mate, when it exists, can be read off directly from a formula rather than found by search. A near-factorization is a pair of subsets (A,B) whose pairwise products cover every nonidentity group element exactly once. Writing M(A) for the |G|×|G| incidence matrix of A, the paper shows the mate's incidence matrix must equal (1/|A|)J - M(A)^{-1}, where J is the all-ones matrix; this identity both proves uniqueness and makes the mate computable by one matrix inversion or one sparse linear solve. The authors then use this machinery to rule out nontrivial near-factorizations in every noncyclic abelian group of order below 200, to give a short proof that a strong circular external difference family (a collection of disjoint subsets with prescribed differences between consecutive sets) has at most two sets, and to exhibit index-2 near-factorizations in some noncyclic abelian groups.

What carries the argument

The load-bearing object is the incidence matrix M(H) defined by M(H)_{i,j}=1 when $g_i^{{-1}}$g_j is in H. The key identity is the characterization M(A)M(B)=J-I for near-factorizations, which the paper combines with the row-sum identity M(A)J=|A|J to force the unique formula M(B)=(1/|A|)J-M(A)^{-1}. All later results, including the nonexistence of strong circular external difference families with more than two sets and the exhaustive nonexistence searches, hang on this matrix identity.

What would settle it

Apply Algorithm 2 to the three groups absent from Table 1, namely Z8×(Z2)^3, (Z4)^2×(Z2)^2, and Z4×(Z5)^2. If any symmetric candidate set A yields a 0-1 solution z to M(A)z=(0,1,...,1)^T with |A| times the Hamming weight of z equal to |G|-1, then a nontrivial near-factorization exists and the paper's exhaustive nonexistence claim for order below 200 fails.

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Extended reading notes

Core claim

For a finite group G with identity e, subsets A and B form a near-factorization when |A||B|=|G|-1 and the products ab with a in A and b in B are exactly G\{e}. The central discovery is that the mate is unique: if M(A)M(B)=J-I encodes the near-factorization condition, then invertibility of M(A), which has constant row sum |A|, forces M(B)=(1/|A|)J-M(A)^{-1}. The paper presents two algorithms based on this identity: one computes the explicit inverse, and the other solves the sparse linear system M(A)z=(0,1,...,1)^T and recovers B from the positions where z has a 1. Using these algorithms together with structural reductions, all noncyclic abelian groups of order less than 200 are checked for nontrivial near-factorizations, and none is found.

Load-bearing premise

The exhaustive claim that no noncyclic abelian group of order below 200 has a nontrivial near-factorization rests on the completeness of Table 1; as printed, that table omits the order-64 groups Z8×(Z2)^3 and (Z4)^2×(Z2)^2 and the order-100 group Z4×(Z5)^2, so the conclusion only holds if those groups are covered by cited theorems or performed searches.

Editorial extensions

If this is right

If a set A has a mate in a near-factorization, that mate is unique; no second set can realize the same A.
Computing the mate becomes a direct algebraic computation rather than a search over all possible subsets B of the right size.
The uniqueness statement yields a short proof that a strong circular external difference family cannot contain more than two sets.
No noncyclic abelian group of order less than 200 admits a nontrivial near-factorization, according to the paper's theoretical and computer-assisted enumeration.
Noncyclic abelian groups do admit near-factorizations with index lambda=2, with several explicit examples of the form (4,s) listed in the paper.

Reading between the lines

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

If Table 1 is truly exhaustive, the natural conjecture is that nontrivial near-factorizations do not exist in noncyclic abelian groups; the index-2 examples show the analogous statement for lambda>1 is false.
The same matrix formula likely extends to index lambda>1, giving M(B)=(lambda/|A|)J-M(A)^{-1} whenever M(A) is invertible, which would make systematic searches for higher-index examples inexpensive.
The three group orders missing from the printed Table 1 are the first thing to check before relying on the 'all order below 200' conclusion; if any of those groups admits a near-factorization, the exhaustive claim is incomplete.

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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

Kreher, Martin, and Stinson give a clean linear-algebra characterization of near-factorizations: if X=M(A), then any mate satisfies M(B)=(1/r)J - X^{-1}. So a mate is unique, and you can compute it by inverting one sparse matrix. They are upfront (Remark 1.1) that uniqueness follows from de Caen et al.'s proof; the contribution is the explicit formula and the resulting algorithm. That is the genuinely new and useful part.

The paper also does several things well. Algorithm 2 turns the mate search into solving a sparse linear system, a real speedup over brute-force search for B. The alternate proof of the SCEDF nonexistence result is short and transparent. The structural analysis for Z_t × (Z2)^2 is a nice reduction, and the index-2 examples in Table 3 are concrete and checkable.

The soft spots are mostly about reproducibility. The headline claim—no nontrivial near-factorization in a noncyclic abelian group of order < 200—leans on several exhaustive computer searches (n = 116, 136, 148, 153, 156, 171, 172, 175, 176, 188, 196). The paper gives Algorithm 2 and some orbit reductions, but no code, no search counts, no logs. For some of those groups, e.g. Z17 × (Z3)^2, the structural reduction of Section 2.2 doesn't apply, so the search is essentially brute force over symmetric A. That is a real gap: a single pruning mistake would invalidate the headline. The authors should provide code or at least detailed search statistics.

The reader's worry about Table 1 omitting groups of orders 64 and 100 is less serious than it looks. The omitted groups are covered by the stated theorems: Z8 × (Z2)^3 and (Z4)^2 × (Z2)^2 are ruled out via quotient (Z2)^4/Theorem 3.3 or Theorem 3.4, and Z4 × (Z5)^2 is ruled out by Theorem 3.5 with p=5, m=2. So that is a minor presentational issue, not a flaw.

The math that is actually written down is correct and clearly derived. The paper deserves a serious referee. I would send it out, but ask the authors to make the computational part verifiable—code or logs—before accepting.

Referee Report

1 major / 4 minor

Summary. The paper studies near-factorizations (A,B) of finite groups, where AB = G \ {e} and |A||B| = |G|-1. Its main theoretical result is Theorem 1.5: the mate B of a given set A is unique; this follows from the matrix formula M(B) = (1/r)J - M(A)^{-1} (Theorem 1.4). The authors use this to give Algorithm 2, which computes a mate by solving one sparse linear system instead of searching over all candidates B. They then give an alternative proof of a theorem of Wu, Yang and Feng that strong circular external difference families cannot have more than two sets, by applying the uniqueness theorem. They prove structural constraints on near-factorizations in groups of the form Z_t × (Z2)^2, and they report a complete case analysis for all noncyclic abelian groups of order less than 200, claiming that no nontrivial near-factorization exists in any of them, using a combination of earlier theorems and exhaustive computer searches. Finally, they exhibit several index-2 near-factorizations in noncyclic abelian groups.

Significance. If correct, this is a useful contribution to the study of near-factorizations. The uniqueness theorem (Theorem 1.5) and the explicit formula (Theorem 1.4) are clean and conceptually important, because they turn a combinatorial search for a mate into a single matrix inversion or sparse linear solve; this is a genuinely valuable algorithmic observation. The alternate proof of the SCEDF nonexistence theorem is elegant and more direct than the group-ring proof in the literature. The structural analysis for Z_t × (Z2)^2 and the systematic table of nonexistence results for all noncyclic abelian groups below order 200, if fully verified, substantially extend the known nonexistence range from order 100 to order 200. However, a significant part of the central nonexistence claim rests on computer searches whose details are not given, so the paper's contribution is stronger as a theoretical/methodological one than as a fully checkable computational result.

major comments (1)

[Section 3, Table 1] The authority 'exhaustive computer search' is the sole basis for ruling out the parameter sets listed for n = 116, 136 (two rows), 148, 153, 156, 171, 172, 175, 176, 188, and 196 (two rows). The manuscript gives no code, no scripts, no search counts, no orbit representatives, and no description of the pruning rules beyond the partial analysis in Sections 2.1 and 2.2. For example, the n = 153 row (G = Z17 × (Z3)^2, (r,s) = (8,19)) is not covered by the structural reduction of Section 2.2, so it is not clear how the search space was reduced to a verifiable size. Since these searches are load-bearing for the headline claim that no nontrivial near-factorization exists in any noncyclic abelian group of order less than 200, the paper should supply a complete, reproducible description of each search: the exact enumeration strategy, the number of candidate A-sets (or orbit representatives) tested, and the verification procedure. Without this, the central computational conclusion cannot be independently checked.

minor comments (4)

[Section 3, Table 1] The sentence 'All the noncyclic abelian groups of order at most 200 are listed in Table 1' is not accurate as printed: the table omits Z8 × (Z2)^3 and (Z4)^2 × (Z2)^2 of order 64 and Z4 × (Z5)^2 of order 100. These groups are in fact covered by the cited theorems (Theorem 3.3 for Z8 × (Z2)^3, Theorem 3.4 for (Z4)^2 × (Z2)^2, and Theorem 3.5 for Z4 × (Z5)^2), so the nonexistence conclusion is not affected, but the table's completeness statement and the table itself need to be corrected.
[Section 2.2] The assertion near equations (5)-(8) that these equations are satisfied 'if and only if' the exceptional index for which a_{i,j} = (r+3)/4 coincides with the exceptional index for which b_{i,j} = (s-3)/4 is stated without proof; since this fact is used to accelerate the search in Example 2.1, a short verification should be supplied.
[Section 5] The sentence 'we have not found any (r,s,2)-near-factorizations in noncyclic abelian groups of even order' is imprecise: the paper should state the exact groups and parameter ranges that were searched and whether those searches were exhaustive, so the reader can interpret this observation correctly.
[Section 6] The section heading contains a typo: 'Discusion' should be 'Discussion'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the unique-mate formula is derived algebraically from a prior matrix characterization, and the computational searches are empirical claims, not fitted predictions.

full rationale

The derivation chain is non-circular. Theorem 1.5 follows from the matrix equation M(A)M(B)=J-I (Theorem 1.3, cited from de Caen et al.) together with Theorem 1.4, whose proof is direct algebraic manipulation: XJ=rJ implies Y=(1/r)J-X^{-1}. No parameter is fitted to data; the mate B is computed from A by solving a linear system, and the 'prediction' of B is not an input to that computation. The alternate SCEDF proof applies Theorem 1.5 to carefully identified near-factorizations, which is a legitimate application rather than a circular dependency. The paper relies heavily on [3], which includes a co-author, but that reliance is on a prior published, parameter-free structural theorem with stated assumptions, not on an unverified self-citation whose conclusion is the present claim. The notable weaknesses—Table 1's omission of some groups of order 64 and 100 (coverable by Theorems 3.3–3.5) and the absence of code or logs for several 'exhaustive computer search' entries—are correctness and reproducibility concerns, not circularity. Accordingly, no circular step is identified.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The paper's theoretical results are derived from a small set of cited matrix-encoding theorems and standard finite group classification. No free parameters are fitted; no new entities are postulated. The main unstated assumption is the completeness of the table and the correctness of the ad hoc computer searches, which are captured as red flags rather than as formal axioms.

assumptions (3)

domain assumption Theorem 1.3 of de Caen et al.: (A,B) is a near-factorization iff M(A)M(B) = J - I over the integers.
This matrix equivalence is invoked at the start (Theorem 1.3) and underlies the uniqueness formula; it is cited, not proved.
domain assumption Lemma 1.2 of de Caen et al.: in an abelian group, any (r,s)-near-factorization gives a symmetric (r,s)-near-factorization.
The exhaustive searches in Section 3 and the structural analysis in Section 2.2 restrict to symmetric A and B; the whole search design depends on this lemma.
standard math Classification of finite abelian groups: every finite abelian group is a product of cyclic groups of prime-power order, and the invariant factors determine the group.
The enumeration in Table 1 and the calculations for order 144 assume this classification without proof.

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Pith. "Pith review of Uniqueness and explicit computation of mates in near-factorizations." pith.science (2026). https://pith.science/paper/2NZEWTJQ

@misc{pith2026241115890,
  author       = {Pith},
  title        = {Pith review of: Uniqueness and explicit computation of mates in near-factorizations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2NZEWTJQ}},
  note         = {Machine review of arXiv:2411.15890}
}

abstract

We show that a "mate'' $B$ of a set $A$ in a near-factorization $(A,B)$ of a finite group $G$ is unique. Further, we describe how to compute the mate $B$ very efficiently using an explicit formula for $B$. We use this approach to give an alternate proof of a theorem of Wu, Yang and Feng, which states that a strong circular external difference family cannot have more than two sets. We prove some new structural properties of near-factorizations in certain classes of groups. Then we examine all the noncyclic abelian groups of order less than $200$ in a search for a possible nontrivial near-factorization. All of these possibilities are ruled out, either by theoretical criteria or by exhaustive computer searches. (In contrast, near-factorizations in cyclic or dihedral groups are known to exist by previous results.) We also look briefly at nontrivial near-factorizations of index $\lambda > 1$ in noncyclic abelian groups. Various examples are found with $\lambda = 2$ by computer.

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Forward citations

Cited by 1 Pith paper

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Reference graph

Works this paper leans on

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

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Pith tools

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

[1] [1]

Bacs´ o, L

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

work page 2008

[2] [2]

De Bruijn

N.G. De Bruijn. On number systems Nieuw Arch. Wisk. 3 (1956), 15–17

work page 1956

[3] [3]

de Caen, D.A

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

work page 1990

[4] [4]

Grinstead

C.M. Grinstead. On circular critical graphs. Discr. Math. 51 (1984), 11–24

work page 1984

[5] [5]

Hillar and D.L

C.J. Hillar and D.L. Rhea. Automorphisms of ﬁnite abelian groups. Amer. Math. Monthly 114 (2007), 917–923. 16

work page 2007

[6] [6]

Huczynska and M.B

S. Huczynska and M.B. Paterson. Existence and non-existence results for strong external dif- ference families. Discr. Math. 341 (2018), 87–95

work page 2018

[7] [7]

Strong External Difference Families and Classification of $\alpha$-valuations

D.L. Kreher, M.B. Paterson and D.R. Stinson. Strong external d iﬀerence families and classiﬁ- cation of α-valuations. Preprint. https://arxiv.org/abs/2406.09075

work page Pith review arXiv

[8] [8]

Paterson and D.R

M.B. Paterson and D.R. Stinson. Circular external diﬀerence fam ilies, graceful labellings and cyclotomy. Discr. Math. 347 (2024), article 114103, 15 pp

work page 2024