Affine Equivalence of Subsets of $\mathbb{F}_2^n$ via Venn Diagrams and Applications to Sidon Sets

Daniel Rose-Levine; Kariane Calta; Lauren L. Rose; Sarah Covey; Timothy E. Goldberg

REVIEW 3 major objections 3 minor 15 references

Affine Equivalence of Subsets of $\mathbb{F}_2^n$ via Venn Diagrams and Applications to Sidon Sets

T0 review · 3 major / 3 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read Two subsets of F_2^n are affinely equivalent exactly when their even zero-sum spaces yield cardinality-preserving linearly isomorphic Venn region structures.

desk verdict Main affine-equivalence criterion is sound and useful, but there is a false change-of-basis theorem and Section 7 leans on an unproven external enumeration. read the letter →

arxiv 2509.00556 v1 pith:5NS2CLEU submitted 2025-08-30 math.CO

Kariane Calta , Sarah Covey , Timothy E. Goldberg , Lauren L. Rose , Daniel Rose-Levine This is my paper · ORCID

classification math.CO MSC 05B2511B3051E15

keywords affineequivalenceVenndiagramsevenzero-sumsetsSidoncapsfinitegeometrybinaryvectorspacesclassification

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 proves that affine equivalence of subsets of F_2^n can be detected from the Venn diagram of a naturally defined linear space: the collection E(S) of even-size subsets of S that sum to zero. The main theorem states that S and T are affinely equivalent if and only if there is a cardinality-preserving linear bijection between their Venn region spaces. This turns a search for an affine automorphism into a finite linear-algebra comparison. The paper then uses this criterion to classify Sidon sets (caps): it fully classifies caps whose size exceeds dimension by 3, and it verifies the affine equivalence classes of 10-, 11-, and 12-caps in dimension 7.

What carries the argument

The even zero-sum space E(S): the F_2-linear subspace of P(S) (with symmetric difference as addition) containing all even subsets of S that sum to zero in F_2^n. Since affine combinations in characteristic 2 are exactly odd sums, E(S) captures affine dependence, and affine maps are precisely bijections preserving it. The Venn diagram of E(S) is made into an F_2-vector space whose regions are indexed by coordinate vectors; a change of basis in E(S) transforms the region coordinates by the transpose-inverse matrix, so the vector-space structure of the regions is basis-independent. The proof then equates (V,W)-isomorphisms of sets with cardinality-preserving linear bijections of the correspondi

What would settle it

An exhaustive computer enumeration over all subsets of F_2^n for n up to 6 (or over all 7-dimensional caps) checking whether every pair of sets is affinely equivalent if and only if a cardinality-preserving linear bijection exists between their Venn region spaces; a single pair with a matching cardinality-preserving linear bijection but no affine equivalence, or vice versa, would disprove Theorem 5.7, and a 7-dimensional cap matching no template from [2] would falsify the application's completeness assumption.

Watch

Extended reading notes

Core claim

The central claim is Theorem 5.7: for subsets S,T of F_2^n, there is an affine automorphism taking S to T if and only if there is a cardinality-preserving linear bijection from the Venn diagram of E(S) to the Venn diagram of E(T), where E(S) is the subspace of the power set P(S) consisting of even-cardinality subsets whose elements sum to zero. The discovery is that affine structure is completely encoded by which even subsets sum to zero, and that this algebraic data can be reorganized into a Venn diagram that is independent of the chosen basis. The proof shows that any bijection preserving E(S) extends to an affine map, and that such bijections are exactly the cardinality-preserving linear

Load-bearing premise

The Section 7 classification assumes that the extended cap basis type templates listed in the companion paper [2] include every possible 10-, 11-, and 12-cap of dimension 7; if some cap fits none of those templates, the conclusions that all 11-caps are equivalent (or that the 10-cap classes split as described) could be wrong.

Editorial extensions

If this is right

Affine equivalence of any pair of subsets of F_2^n can be decided by comparing the linear structure of Venn region cardinalities, without constructing an explicit affine map.
When the Venn region cardinality multisets differ, the two sets are certainly not affinely equivalent, giving a cheap negative test (Corollary 4.4).
Caps with size minu dimension equal to 3 are classified by two data: the multiset of Venn region cardinalities and the number of isolated points, leading to an explicit parameterization and machine count of equivalence classes up to k=27.
For the 7-dimensional caps carried over from the companion template list, the method reproduces the known classes: 10-caps split into two equivalence classes, while all 11-caps and all 12-caps in the templates are equivalent.
The main theorem is fully general, applying to all subsets—not only caps—so the Venn criterion is a candidate tool for classifying Sidon sets in higher dimensions where complete classification remains open.
The approach yields a canonical linear invariant, E(S), of dimension |S| minus (dim(S)+1), computed directly from any affine basis of S, so no search over automorphisms is needed to state the invariant.

Reading between the lines

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

A natural next step is to test the criterion computationally for n between 8 and 10, where maximum Sidon set sizes are still open, by comparing the Venn cardinality linear structures as an equivalence invariant against known constructions.
The method may be recast as classifying subsets by the isomorphism type of a weighted linear code—the even zero-sum space together with region weights—which could connect the classification to coding-theoretic invariants such as hulls or weight enumerators.
The central mechanism depends essentially on characteristic 2, since affine combinations reduce to parity; an open question is whether an analogous 'weight-restricted zero-sum subspace' invariant characterizes affine equivalence over F_q for odd q.
The Section 7 conclusions inherit an external completeness assumption about the template list; an independent exhaustive enumeration of 7-dimensional caps up to affine equivalence would either confirm or refute the 'all 11-caps equivalent' and 'all 12-caps equivalent' statements.

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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

The core idea is good: affine equivalence of subsets of F2^n can be read off from the linear structure of Venn region cardinalities, and Theorem 5.7 is likely the right way to frame this. Section 6's classification of k-caps with dimension k-3 is clean, concrete, and easy to reproduce; that is a genuine new contribution worth crediting.

But there is a real bug in the middle. Theorem 3.16 claims that under a change of basis X -> Y of V, the Venn coordinates transform by (M^T)^{-1}. That is false. Take S={1,2,3}, V=span({1},{1,2}), bases X=({1},{1,2}) and Y=({1},{2}). The change-of-basis matrix M is [[1,0],[1,1]], but the coordinate change on Venn regions is multiplication by M, not by (M^T)^{-1}=[[1,1],[0,1]]. The proof breaks because it assumes every basis vector has a nonempty exclusive region, which need not hold. Corollary 3.14 also overstates: a nonempty X in V need not contain exactly 2^{r-1} nonempty regions.

Good news: these errors do not sink Theorem 5.7. Theorem 4.5's converse is proved without the faulty parts, and the basis-independence of addition on Venn regions actually follows directly because v_Y^{-1}∘v_X is always linear. So the main criterion appears sound. Still, the paper as written contains an incorrect theorem and needs revision.

The second soft spot is Section 7. The classification of 10-, 11-, and 12-caps in dimension 7 assumes the templates from the companion paper [2] are exhaustive. That completeness is not proved or checked here; if a template is missing, the equivalence-class conclusions would be wrong. This is an external premise, not an error in the Venn method, but it should be flagged prominently.

Also, Theorem 2.6 — affine equivalence iff there is an even-zero-sum-preserving bijection — is a standard matroid fact. The authors should credit prior work on vector matroids over F2 rather than present it as new.

Who is this for? Anyone working on affine equivalence in F2^n, Sidon sets, or caps. The method is transparent, and Section 6 gives a solid hook. I would send it to a serious referee, with a specific request to check Theorem 3.16 and the completeness assumption in Section 7. Fix those, and it becomes a useful paper.

Referee Report

3 major / 3 minor

Summary. The paper develops a Venn-diagram invariant for subsets S of F2^n relative to a linear subspace V of the power set P(S). For a basis of V it partitions S into Venn regions and introduces a vector-space structure on the collection of regions. It proves that this partition and the vector-space structure are independent of the basis of V (Theorem 3.16, Corollary 3.17), and that a bijection S -> T is an affine equivalence exactly when it induces an isomorphism of the even-zero-sum subspaces E(S), E(T) (Theorems 2.6 and 5.6). The main criterion (Theorem 5.7) states that S and T are affinely equivalent iff there is a cardinality-preserving linear bijection between Venn(S,E(S)) and Venn(T,E(T)). The paper applies this to caps (Sidon sets): Section 6 gives a complete classification for caps with size-dimension difference 3, and Section 7 uses templates from [2] to classify 10-, 11-, and 12-caps in dimension 7.

Significance. If the proofs are repaired, the main theorem is a valuable and elegant reduction: affine equivalence of arbitrary subsets of F2^n becomes a finite, parameter-free linear-algebra problem on the even-zero-sum subspace. The paper is explicit and the constructions are concrete, with worked examples and tables. The Section 6 characterization of caps with size-dimension difference 3 is complete and constructive, and the Venn-diagram method demonstrably recovers the known 7-dimensional cap classifications. However, the Section 7 application relies on an external enumeration from the companion paper [2], and the proof of Theorem 3.16 contains a false nonemptiness assertion; both issues need attention before the paper is accepted.

major comments (3)

[Section 7, Tables 4, 5, 7] The application to 7-dimensional caps depends on the assertion that the templates reproduced from [2, Sections 5–7] 'cover all possibilities' for 10-, 11-, and 12-caps. No proof of this exhaustiveness is given here. Consequently the conclusions that all 11-caps and all 12-caps of dimension 7 are affinely equivalent, and that the 10-caps split into exactly two classes, are conditional on the completeness of the external enumeration. This does not affect Theorem 5.7, but it is load-bearing for the stated Sidon-set application. Please either supply a proof of exhaustiveness or explicitly state the reliance on [2] with precise pointers to the relevant theorems.
[Theorem 3.16] The proof asserts 'every basis vector e_i is nonzero and has v_X(e_i) nonempty'. This is false in general: for S={1,2,3} and V=span({1},{1,2}) with basis X1={1}, X2={1,2}, one has v_X(10)=X1∩X2^c=∅. Thus Lemma 3.13 cannot be applied to these e_i. Since Corollary 3.17 and Theorem 5.7 rely on the coordinate-change formula, the proof needs repair. The statement may be salvageable by choosing a basis of F2^r consisting of vectors whose Venn regions are nonempty (such a basis exists because the distinct incidence columns span F2^r), but the manuscript must supply a correct argument.
[Lemma 6.2 / Theorem 6.3] In Lemma 6.2 proof, 'each of a + b ≤ 6' should be '≥ 6'; the same typo appears in Theorem 6.3 proof. Also, 'a, b, and c must have the same cardinality' should be 'same parity'. These are presentation typos in a proof, but they obscure an otherwise correct argument and should be fixed.

minor comments (3)

[Example 3.18] In the list of Venn regions, 'v_X(001) = v_Y(111) = {a4}' should likely read 'v_X(111) = v_Y(111) = {a4}', since v_X(001) was already listed as {w3}.
[Section 7.2] The sentence 'Figure 5 shows the Venn cardinality diagrams for the three templates in Table 4' should refer to Table 5, not Table 4.
[Throughout Section 7] The matrices A and B in Sections 7.2 and 7.3 are asserted to permute cardinalities 'appropriately' without a calculation. A short verification, or a reference to an appendix or supplementary file, would make the application easier to check.

Circularity Check

0 steps flagged · score 2.0 of 10

Theorem 5.7 is self-contained; only caveat is §7 reliance on template exhaustiveness from [2], an external completeness assumption rather than a circular derivation.

full rationale

The main derivation chain is not circular. E(S) is defined independently in Definition 2.2 from the affine-geometric notion of even zero-sum sets, and Proposition 5.1 plus Corollary 5.4 prove it is a linear subspace with an explicit basis derived from an affine basis of S. Theorem 2.6 proves affine equivalence iff there is a bijection S→T inducing a bijection E(S)→E(T), using only the standard characterization of affine maps in F2^n and a direct construction of the affine automorphism. Theorem 4.5 proves that a (V,W)-isomorphism exists iff there is a cardinality-preserving linear bijection of Venn regions; this is established by explicit map constructions (Proposition 4.3 and the converse in Theorem 4.5) using Lemma 3.13 and Proposition 3.15, with no fitted parameters or data-driven assumptions. The final criterion Theorem 5.7 therefore follows from Definition 2.2, Proposition 5.1, Corollary 5.4, Theorem 2.6, and Theorem 4.5 without importing any prior classification. The only external dependency is in Section 7, where the author-provided templates from [2] are imported as exhaustive for 10-, 11-, and 12-caps of dimension 7. That exhaustiveness is not proved in this paper and is load-bearing for the cap-classification conclusions stated there. However, this is a normal citation to prior work by overlapping authors, and it does not feed back into Theorem 5.7 or the earlier theory. It is an external completeness premise, not a circular reduction or a fitted prediction. Hence the central claim has independent content, and the paper receives a low non-circularity score.

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

No free parameters are fitted; the only input is the set S itself. The axioms are standard linear algebra over F2 plus one imported completeness assumption from prior work.

assumptions (4)

standard math P(S) is a vector space over F2 under symmetric difference, with singleton subsets as basis
Invoked in Section 3, Proposition 3.3, and throughout to treat Venn regions as vectors.
standard math Every affinely independent set in F_2^n extends to an affine basis of F_2^n
Used in Theorem 2.6 proof to construct the affine automorphism f from a basis extension.
standard math Affine combinations in F_2^n are exactly sums of an odd number of points
Used in Section 2 and Proposition 2.5 to characterize affine maps via even zero-sums.
domain assumption The template lists of [2] exhaustively cover 7-dimensional 10-, 11-, and 12-caps
Imported from the authors' prior work to make the Section 7 classification conclusions; not proven in this paper.

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Pith. "Pith review of Affine Equivalence of Subsets of $\mathbb{F}_2^n$ via Venn Diagrams and Applications to Sidon Sets." pith.science (2026). https://pith.science/paper/5NS2CLEU

@misc{pith2026250900556,
  author       = {Pith},
  title        = {Pith review of: Affine Equivalence of Subsets of $\mathbbF_2^n$ via Venn Diagrams and Applications to Sidon Sets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5NS2CLEU}},
  note         = {Machine review of arXiv:2509.00556}
}

abstract

Two subsets $S$ and $T$ of $\mathbb{F}_2^n$ are \textit{affinely equivalent} if there is an affine automorphism of $\mathbb{F}_2^n$ taking $S$ to $T$. Given a basis of the affine span of $S$, we can construct a Venn diagram whose regions partition $S$. We prove that any two bases of $\operatorname{aff}(S)$ will have the same Venn diagram up to a linear permutation of the Venn regions. Moreover, we prove that two sets are affinely equivalent if and only if there is a cardinality-preserving linear permutation from the Venn regions of $S$ to the Venn regions of $T$. We use these results to classify certain Sidon sets up to affine equivalence.

Figures

Figures reproduced from arXiv: 2509.00556 by the authors.

**Figure 1.** Labels of Venn regions by bitstrings for Venn diagrams of two, three, and four sets. with X1 △ X2 = {w1, w2, a1, a2, a3, a8, a9, a10}, X1 △ X3 = {w1, w3, a1, a2, a5, a6, a7, a8}, X2 △ X3 = {w2, w3, a3, a5, a6, a7, a9, a10}, X1 △ X2 △ X3 = {w1, w2, w3, a1, a2, a4, a9, a10} [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗

**Figure 2.** The Venn diagrams corresponding to the bases X and Y for E(S) in Example 3.9. (2) The function vX : F r 2 → VennX (S, V) is bijective and makes VennX (S, V) into a vector space over F2 of dimension r with addition defined by vX (a) + vX (b) := vX (a + b) for all vX (a), vX (b) ∈ VennX (S, V). (3) The Venn region vX (0) is the set of isolated points for V, so vX (0) = R∅ [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

**Figure 3.** The Venn cardinality diagrams for S1, S2, and S3 from Example 5.8. where w1 = a1 + · · · + a7, w2 = a4 + · · · + a10, w3 = a3 + a4 + a8, x1 = a1 + a2 + a3 + a7 + a8 + a9 + a10, x2 = a3 + a4 + a5 + a6 + a7 + a9 + a10, x3 = a1 + a2 + a4 + a7 + a8 + a9 + a10, y1 = a1 + · · · a5, y2 = a4 + a5 + a8 + a9 + a10, y3 = a3 + a4 + a6 + a7 + a8. Put Si = B ⊔ Di for i = 1, 2, 3. In [PITH_FULL_IMAGE:figures/full_fig_p024_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: The Venn cardinality diagrams for three caps of size 10 and dimension 7. In this section, we will consider each cap template from [2] and use the Venn diagram method to determine which are equivalent and which are not. 7.1. 10-caps of dimension 7 [PITH_FULL_IMAGE:figu…

**Figure 5.** Figure 5: The Venn cardinality diagrams for three caps of size 11 and dimension 7. It is straightforward to check that A permutes the Venn regions of a 7-5-5-(4, 4, 3) template cap to the 5-5-5-(3, 3, 3) cap appropriately, and that B permutes those of a 5-5-5-(3, 3, 3) template …

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Works this paper leans on

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Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

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How Many Cards Should You Lay Out in Quad-128: A Classification of Caps in AG(7,2)

Kariane Calta, Timothy E. Goldberg, and Lauren L. Rose. How Many Cards Should You Lay Out in Quad-128: A Classification of Caps in AG(7, 2). submitted. 2025. arXiv: 2501 . 11173 [math.CO]. url: https://arxiv.org/abs/2501.11173. 32 REFERENCES

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doi:10.1016/j.jcta 2019

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On APN Functions Whose Graphs are Maximal Sidon Sets

C. Carlet. “On APN Functions Whose Graphs are Maximal Sidon Sets”. In: Latin 2022: Theoretical Informatics . Vol. 13568. Springer, Cham., 2022. doi: https://doi.org/10.1007/978-3-031-20624- 5_15

work page doi:10.1007/978-3-031-20624- 2022

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How many cards should you lay out in a game of EvenQuads: a detailed study of caps in AG( n, 2)

Julia Crager et al. “How many cards should you lay out in a game of EvenQuads: a detailed study of caps in AG( n, 2)”. In: Matematica 2.2 (2023), pp. 382–419. issn: 2730-9657. doi: 10.1007/s44007-023- 00047-0. url: https://doi.org/10.1007/s44007-023-00047-0

work page doi:10.1007/s44007-023- 2023

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Sidon sets, sum-free sets and linear codes

Ingo Czerwinski and Alexander Pott. “Sidon sets, sum-free sets and linear codes”. In: Advances in Mathematics of Communications 18.2 (2024), pp. 549–566. issn: 1930-5346. doi: 10 . 3934 / amc . 2023054. url: https://www.aimsciences.org/article/id/65644c12eca2737fc4ef4e2f

work page 2024

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New constructions of large cyclic subspace codes via Sidon spaces

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work page doi:10.3934/amc.2022074 2024

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Sidon sets, thin sets, and the nonlinearity of vectorial Boolean functions

G´ abor P. Nagy. “Sidon sets, thin sets, and the nonlinearity of vectorial Boolean functions”. In: Journal of Combinatorial Theory, Series A 212 (2025), p. 106001. issn: 0097-3165. doi: https://doi.org/10.1016/ j . jcta . 2024 . 106001. url: https : / / www . sciencedirect . com / science/article/pii/S0097316524001407

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Construction of a cryptographic function based on Bose-type Sidon sets

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work page doi:10.3934/math.2024855 2024