{"id":"259dd594-ac51-479b-8491-51accfe43366","arxiv_id":"2411.15884","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Two known near-factorization constructions for dihedral groups are shown to be equivalent, and new nonequivalent near-factorizations are found in D41, D95, D5*Z5, and C5^2 semidirect C2, including an infinite family.","lead":"This paper studies near-factorizations of dihedral groups, showing that two known construction families are equivalent and giving new examples of non-equivalent near-factorizations, including an infinite family. The work corrects a claim in the published literature and expands the small catalogue of near-factorizations in nonabelian groups, which connects to combinatorial designs used in cryptography.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Computational completeness claims (Thm 2.10, Sections 4.2-4.4) are not independently checkable; without code or algorithm details the uniqueness/nonequivalence conclusions rest on an unverified search.","rationale":"The reader's weakest_assumption identifies the same concern I would raise: the computational completeness and equivalence testing are not independently verifiable from the preprint. The analytic results, especially Theorem 2.4 and the Pecher-transform theorems, appear internally sound, and Theorem 4.1 is built on those arguments plus a cited construction, so I do not see a separate algebraic flaw there. The computational sections are central to the paper's claims about how few nonequivalent near-factorizations exist and about the new nonequivalent examples, so if the unspecified search is wrong, those claims fail. The apparent x/y swap in Section 4.3 reinforces that these sections were not written to the same standard of verifiability as the main theorems. Since this is a reproducibility gap rather than a demonstrated incorrect result, the reader's CONDITIONAL verdict remains appropriate.","tokens_in":19883,"tokens_out":13164,"duration_ms":116838,"concrete_test":"Independently re-run the full equivalence-class enumeration from the definitions, with a fresh implementation or GAP: canonical forms for all (k,(2n-1)/k) near-factorizations in D_n for n <= 32, the (9,21) classes in D_95, and the (7,7) classes in D_5 x Z_5 and C_5^2 semidirect C_2, using the automorphism groups stated in Sections 4.3-4.4. If any count differs from 'unique' or 'exactly two,' the corresponding claim must be weakened; if all counts match, the conditional can be lifted.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Reading in good faith, the analytic core is solid: Theorem 2.4 is an explicit, checkable equivalence via f_{-d1,0}, and the Pecher-transform theorems are carefully argued. The load-bearing gap is the computational part. Theorem 2.10 asserts uniqueness up to equivalence for every n <= 32; Section 4.2 asserts exactly two symmetric (9,21) near-factorizations in Z_190; Section 4.3 asserts exactly two (7,7) classes in D_5 x Z_5; Section 4.4 asserts exactly two in C_5^2 semidirect C_2. In each case the search and equivalence-test algorithm is described only as 'the technique from [9], adapted to dihedral groups,' with no code, output data, or statement of how canonical forms and automorphism groups were computed. Section 4.3's proof of aut(D_5 x Z_5) also appears to swap the roles of f(a) and f(b), saying f(b), the image of order-5 element b, is an order-2 reflection, and Section 4.4 simply lists generators of aut(G) without proof. If any of these computations has a missed equivalence or a false nonequivalence, the claimed scarcity of examples and the specific new examples in those sections would fail. This is an addressable reproducibility gap, not a challenge to the main structural theorems.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":20183,"tokens_out":19407,"duration_ms":163301,"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":[{"comment":"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.","section":"Section 4.1 (Theorem 4.1)"},{"comment":"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":"Theorem 2.10 and Sections 4.2-4.4"},{"comment":"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":"Section 4.3"},{"comment":"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.","section":"Section 3.11 (proof of Theorem 3.11, h=a case)"}],"minor_comments":[{"comment":"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}}.","section":"Theorem 2.4"},{"comment":"The name 'Bocsó' appears in Definition 2.7, while the reference list uses 'Bascó'; please unify the spelling.","section":"Definition 2.7 and references"},{"comment":"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.","section":"References [9], [12]"}],"recommendation":"major_revision","confidential_remarks":"The manuscript's reliance on the authors' own unpublished preprints for enumeration and blowups, together with the unavailability of code or data for the exact computational claims, should be addressed before acceptance. I do not see a circularity problem; this is a reproducibility and proof-completeness issue. The local errors in Sections 3.11 and 4.3 appear fixable without changing the main conclusions, which is why I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nWhat 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.\n\nThe 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.\n\nNone 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.","headline":"Solid algebraic core with a real correction to the literature; computational sections need code or data before the uniqueness claims can be trusted.","tokens_in":20706,"tokens_out":4201,"would_cite":true,"duration_ms":33627,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20D60","05B30"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["near-factorization","dihedral group","strong external difference family","generalized strong external difference family","group equivalence","Pécher transform","computer enumeration","nonequivalent constructions"],"falsifier":"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.","tokens_in":122,"feed_emoji":"🧩","tokens_out":7980,"duration_ms":127634,"temperature":0.7,"pith_summary":"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.","feed_headline":"Dihedral near-factorizations thought distinct are equivalent","feed_subtitle":"An explicit automorphism identifies the two constructions, and new computer examples show when distinctness is real.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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 $D_{95}$."],"supporting_citations":[{"why":"Supplies the 1990 construction of near-factorizations of $D_n$ for all factor pairs, the first family shown equivalent in Theorem 2.4.","marker":"[4]"},{"why":"Supplies the 2008 construction and the assertion of nonequivalence that Theorem 2.4 refutes; also defines the alternating-graph invariant.","marker":"[1]"},{"why":"Supplies the Pécher transform from cyclic to dihedral near-factorizations and the earlier examples in $D_5 \\times \\mathbb{Z}_5$ and $C_5^2 \\rtimes C_2$.","marker":"[13]"},{"why":"Supplies the enumeration technique on which the computer classifications for $n\\le 32$ and the new examples are based.","marker":"[9]"},{"why":"Provides the lemma connecting near-factorizations to two-set generalized strong external difference families, used throughout the discussion.","marker":"[12]"}],"fun_headline_variants":["Automorphism reveals 1990 and 2008 near-factorizations match","Dihedral near-factorizations: two known constructions are equivalent","New non-equivalent near-factorizations in dihedral groups found","Contrary to 2008 claim, dihedral near-factorizations coincide"],"cache_read_input_tokens":22784,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Automorphism reveals 1990 and 2008 near-factorizations match","Dihedral near-factorizations: two known constructions are equivalent","New non-equivalent near-factorizations in dihedral groups found","Contrary to 2008 claim, dihedral near-factorizations coincide"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000297,"raw_usage":{"total_tokens":1683,"prompt_tokens":870,"completion_tokens":813,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":486,"completion_tokens_details":{"reasoning_tokens":739}},"tokens_in":486,"tokens_out":813,"duration_ms":7103,"temperature":1.0,"reasoning_tokens":739,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T13:47:11.185412+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"de Caen, D.A","cited_arxiv_id":null,"evidence_quote":"Supplies the 1990 construction of near-factorizations of $D_n$ for all factor pairs, the first family shown equivalent in Theorem 2.4."},{"cited_title":"Bacs´ o, L.cH´ ethelyi and P","cited_arxiv_id":null,"evidence_quote":"Supplies the 2008 construction and the assertion of nonequivalence that Theorem 2.4 refutes; also defines the alternating-graph invariant."},{"cited_title":"Pˆ echer","cited_arxiv_id":null,"evidence_quote":"Supplies the Pécher transform from cyclic to dihedral near-factorizations and the earlier examples in $D_5 \\times \\mathbb{Z}_5$ and $C_5^2 \\rtimes C_2$."},{"cited_title":"Paterson and D.R","cited_arxiv_id":null,"evidence_quote":"Provides the lemma connecting near-factorizations to two-set generalized strong external difference families, used throughout the discussion."}],"review_version":1}