Enumerations of 1-rotational Steiner systems

Alex Ravsky; Ivan Hetman; Taras Banakh

REVIEW 3 major objections 4 minor 7 references

Enumerations of 1-rotational Steiner systems

T0 review · 3 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read A census of 1-rotational Steiner systems reports 676 non-isomorphic unitals of order 4, versus 67 in an earlier list.

desk verdict Interesting but unverifiable as written: the headline 676 count does not match the paper's own table, and the alleged counterexample to Buratti needs more than "Computer calculations show." read the letter →

arxiv 2505.00469 v1 pith:6F6SARJP submitted 2025-05-01 math.CO

Ivan Hetman , Taras Banakh , Alex Ravsky This is my paper · ORCID

classification math.CO MSC 05B0505B30

keywords Steinersystems1-rotationaldesignsdifferencefamiliesunitaloforder4enumerationnon-isomorphicgroupactionsS(2565)

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

This paper reports a large computer enumeration of $1$-rotational Steiner systems: block designs $S(2,k,v)$ on $v$ points in which a group of order $v-1$ acts regularly on all points except one fixed point. The headline result is a census of $1$-rotational unitals of order $4$, the Steiner systems $S(2,5,65)$: for acting groups of order $64$ the search finds $676$ non-isomorphic designs, in contrast to the $67$ listed in the comparison data [3]. The same search produces seven non-isomorphic designs for $(v,k)=(25,3)$, all realized by the group $\mathrm{SL}(2,3)$, which contradicts a previously published uniqueness theorem for that parameter pair. If the enumeration is correct, the paper substantially revises the known inventory of small $1$-rotational Steiner systems and shows that one design can be generated by several non-isomorphic groups.

What carries the argument

The central object is a $1$-rotational difference family: a list of $k$-subsets (blocks) of a finite group $G$ of order $v-1$ such that the translates of these blocks, together with a formal fixed point $\infty$, form the Steiner system $S(2,k,v)$, with $G$ acting regularly on the non-fixed points. Two families are identified when an isomorphism of $G$ sends one family to a translate of the other (Definition 1.1); this equivalence is coarser than design isomorphism, so the search records both family counts and design counts. The enumeration machinery is an exhaustive computer search over groups of each admissible order, with blocks encoded in a fingerprint notation that records the group action, filtering by multipliers and by the equivalence relation, and finally checking design isomorphism using automorphism-group order. The search algorithm itself is taken from the authors' earlier work [1] and is not restated in the paper.

What would settle it

Independently re-run the exhaustive difference-family search for $S(2,3,25)$ over the group $\mathrm{SL}(2,3)$ of order $24$; if it does not produce exactly seven non-isomorphic designs, the contradiction with [7] fails. Similarly, an independent census over all groups of order $64$ should reproduce exactly $676$ non-isomorphic unitals of order $4$; any other total would refute the paper's main count.

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Extended reading notes

Core claim

For admissible pairs $(v,k)$, the paper enumerates $1$-rotational difference families over groups of order $v-1$, filters the families by the equivalence relation of Definition 1.1 (an isomorphism of the group composed with translation), and, for the unital census, filters further by automorphism-group order and design isomorphism. The central numerical finding is that there are $676$ non-isomorphic $1$-rotational unitals of order $4$ whose automorphism group has order $64$, compared with $67$ in [3]; additional non-isomorphic designs are reported for acting groups of orders $128$, $192$, $256$, $384$, and $768$. For $(v,k)=(25,3)$ the paper lists seven pairwise non-isomorphic designs, all for $\mathrm{SL}(2,3)$, and states that this contradicts the uniqueness theorem proved in [7]. Tables give, group by group, the number of filtered difference families and, where computed, the number of non-isomorphic designs, and the classical unital $S(2,5,65)$ is shown to arise as a $1$-rotational design for three non-isomorphic groups of order $64$.

Load-bearing premise

The load-bearing premise is that the generalized search algorithm, carried over from the authors' earlier work [1] and modified for $1$-rotational actions, is exhaustive and error-free for every group and parameter set listed, even though the algorithm itself is not described and the code is not included in the paper.

Editorial extensions

If this is right

The published uniqueness result for $(v,k)=(25,3)$ is false as stated, because the paper exhibits seven pairwise non-isomorphic $1$-rotational designs for that pair, all for $\mathrm{SL}(2,3)$.
The known count of $1$-rotational unitals of order $4$ must be revised upward: at least $676$ non-isomorphic examples arise from groups of order $64$ alone, against the $67$ in [3].
The classical unital $S(2,5,65)$ is $1$-rotational under at least three non-isomorphic groups of order $64$, so being $1$-rotational is not a property of a unique acting group.
The tables supply many new existence examples for $1$-rotational Steiner systems with block sizes $3$, $4$, and $5$, including affine planes of orders $3$ and $5$ that admit several distinct $1$-rotational realizations.
Because the difference-family equivalence relation of Definition 1.1 is coarser than design isomorphism, the family counts in the tables cannot be read as design counts; the final isomorphism layer is essential.

Reading between the lines

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

An independent implementation of the search is the natural check on the census; until that is done, the exact totals in the tables are best treated as strong computational evidence rather than proven enumerations.
If the $676$ count survives independent verification, earlier counts of $1$-rotational unitals of order $4$ are incomplete by an order of magnitude, and the same discrepancy may affect unitals of higher orders.
The paper leaves implicit that the same machinery, with the isomorphism checking improved, could be pointed at unitals of order $5$ or at larger block sizes; the bottleneck it names is checking design isomorphism, not generating difference families.
The repeated fingerprints among the seven $(25,3)$ designs show that fingerprint data alone cannot decide isomorphism, so any automated use of the raw tables would need a separate isomorphism invariant or check.

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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

First, what is actually in here: a sweep over 1-rotational difference families for small parameters, with the big claims being 676 non-isomorphic 1-rotational unitals of order 4 with automorphism group of order 64 (previous count 67) and seven 1-rotational STS(25)s contradicting a uniqueness theorem from Buratti. If those numbers are right, this is a real expansion and a real correction.

Good things: the scope is sensible, they list explicit difference families, the raw output is placed on GitHub, and they are honest about not having compared with [4]. Their earlier paper [1] gives the underlying algorithm, so this is not a ball of smoke. The examples for k = 3, 4, 5 give concrete families that can be checked, and they account for multiplier filtering and automorphism group order in the unital case.

Now the soft spots.

First and most concrete, the enumeration in Section 4 has an arithmetic inconsistency. Summing the "non-iso designs" column for SmallGroup(64,·) gives 967, while the prose says 676 designs with automorphism group of order 64. There is no stated global deduplication or aut-group-order filter that turns one into the other. Maybe the table lists all non-isomorphic designs found per group while the 676 are those with aut group exactly 64, but the reader should not have to guess.

Second, the algorithm is not described. The paper says it is a "very easily generalized" version of [1] and that "corresponding changes were applied," but there is no description of the search space, the difference-family representation, the isomorphism check, or a proof of exhaustiveness. Without code or a precise specification, the counts are only as good as that trust. The GitHub link is to raw results, not code or proof.

Third, Example 2.4 needs a much better justification. The claim that seven designs are non-isomorphic is supported by "Computer calculations show" with no canonical-label certificate or description of the isomorphism test. Some of the seven share the same fingerprints, so the check is not trivial. Also, the paper asserts a contradiction to Buratti's theorem without quoting the theorem or explaining which hypothesis is violated. It may well be right, but as it stands it is a pointer, not a counterexample.

Where this lands: it is a computational research note with potentially useful data, but the paper does not yet provide enough for the data to be used with confidence. If I were handling it, I would send it to a referee, mainly because the alleged counterexample to a published theorem is important enough to sort out. The revision needs code, a clear algorithm statement, an explanation of the 676/967 discrepancy, and a direct quote and analysis of [7]. For a specialist in designs with computational leanings, this is worth reading. I would not cite it until it is fixed.

Referee Report

3 major / 4 minor

Summary. The paper reports enumerations of 1-rotational Steiner systems S(2,k,v) for several parameter sets, obtained by a group-action generalization of a cyclic difference family search algorithm from the authors' earlier work. The main claimed census is in Section 4: 676 non-isomorphic 1-rotational unitals of order 4, i.e. S(2,5,65), with automorphism group of order 64, together with smaller counts for groups of orders 128, 192, 256, 384, and 768. The paper also lists difference families for k=3, 4, and 5, gives non-existence cases, and in Example 2.4 asserts that there are seven pairwise non-isomorphic 1-rotational Steiner triple systems on 25 points, contradicting a uniqueness theorem of Buratti [7].

Significance. If the enumeration is correct, the paper is significant: it substantially enlarges the known supply of 1-rotational Steiner systems, gives a concrete counterexample to a published uniqueness theorem for STS(25), and corrects the census in [3] for 1-rotational unitals of order 4. The manuscript is also honest about the limits of its comparison with [4], and it makes raw results available through a GitHub repository. However, the numerical conclusions are computational in nature, and the paper as written does not provide enough information to verify the computations: no code, no algorithm description, no isomorphism certificates, and the headline count 676 is not directly recoverable from the table in Section 4. The significance is therefore conditional on the missing computational evidence.

major comments (3)

The text states that '676 non-isomorphic designs with automorphism group of order 64 were found', but the adjacent table's '# of non-iso designs' column sums to 967 over the 36 listed SmallGroup(64,·) rows. The paper never explains the relationship between these two numbers. In particular, it does not describe a global deduplication step across different groups, even though Example 4.1 already exhibits one design (the classical unital) that appears in three different rows (groups 11, 28, and 245), so a simple sum of per-group columns overcounts designs. Nor does it state how the 'automorphism group of order 64' filter was applied or how the automorphism group was computed. Without an explicit reconciliation, the headline census in Section 4 cannot be audited from the paper itself; this is a load-bearing omission for the central claim.
The paper claims seven pairwise non-isomorphic 1-rotational STS(25) and a contradiction to Buratti's uniqueness theorem in [7], but the only support is the sentence 'Computer calculations show that all 7 obtained designs are non-isomorphic'. No isomorphism test, canonical-label computation, or independent certificate is supplied. The need for evidence is especially acute because several of the listed designs have identical fingerprints (designs 1 and 7 both have fingerprint {0=192, 1=13008}; designs 2, 3, and 5 all have {0=1536, 1=11664}), so the claimed non-isomorphism is not evident from the data shown. Since this is the paper's only explicit contradiction of a published theorem, a reproducible verification of pairwise non-isomorphism is required before this claim can be accepted.
All enumeration results depend on a generalized difference-family search algorithm, but the paper does not describe that algorithm. The introduction says the cyclic/commutative/any-group algorithms 'can be very easily generalized' to 1-rotational designs and that 'corresponding changes were applied', without giving pseudocode, a description of the search space, pruning rules, or a proof of exhaustiveness. Section 6 points to a GitHub repository with raw results, but no code or detailed documentation is included in the manuscript. Consequently every table and non-existence claim rests on an unstated computational premise; the reader cannot distinguish an exhaustive census from a partial search. The authors should provide the actual search code, its version, and instructions, or at minimum a complete and precise algorithmic description sufficient for independent reimplementation.

minor comments (4)

The GAP IDs in the k=5 table appear to be inconsistent with the stated group orders: the table lists SmallGroup(25,1), SmallGroup(25,2), and SmallGroup(25,3) for groups C3:C8, C24, and SL(2,3), which all have order 24, so the IDs should presumably start with SmallGroup(24,·). Please correct these identifiers or explain the notation.
There are several typographical errors, including 'occured' in the introduction, 'interse ct' in Section 1, and 'paramodiﬁed unitals' in reference [4]. A careful proofreading pass is needed.
The comparison with [3] would be more informative if the authors reported the exact overlap of their 676 designs with the 67 unitals listed in [3], rather than only the total count and the statement that the new number is bigger.
The paper uses difference-family equivalence as a filter and notes that it is coarser than design isomorphism. For tables in Sections 2, 3, and 5, it is not always clear whether the reported counts are numbers of difference families or numbers of pairwise non-isomorphic designs; the terminology should be made uniform, and the filtering performed for each section should be stated explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the 1-rotational Steiner system counts are algorithmic search outputs, not consequences of the claims being made.

full rationale

The paper's derivation chain is a computational enumeration: an algorithm from the authors' earlier work [1] is generalized from cyclic/commutative groups to arbitrary group actions and applied to 1-rotational difference families. The target counts, including the 676 claimed 1-rotational unitals of order 4 and the seven non-isomorphic S(2,3,25) designs in Example 2.4, are outputs of that search, not inputs. No parameter is fitted to a subset of the enumeration and then renamed as a prediction; no equation defining the result in terms of itself appears. The equivalence relation in Definition 1.1 is a standard notion of difference-family equivalence and is not defined in terms of the final design counts. The contradiction with Buratti's theorem [7] rests on the computational claim that all seven designs are non-isomorphic, which the paper explicitly flags as nontrivial because some have identical fingerprints; this is a correctness/reproducibility concern, not circularity. Similarly, the discrepancy between the sum of the table's 'non-iso designs' column and the stated 676 total is an auditing issue about an undocumented deduplication or automorphism-group filter, not a circular step. The self-citation to [1] is methodological: it supplies the search procedure, not the conclusion that the procedure's outputs are correct. Since no fitted parameter, self-referential definition, or uniqueness theorem imported from the authors' own prior work is used to force the claimed enumerations, the paper has no significant circularity.

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

No free parameters are fitted; this is an exhaustive search census, not a model with tunable constants. The entire edifice rests on computational assumptions: the search is exhaustive, the isomorphism filtering is correct at least for the advertised unital counts, and the GAP/LOOPS tables are faithful. No new entities are introduced.

assumptions (3)

domain assumption The Hetman [1] difference-family search algorithm, generalized to arbitrary group actions, is exhaustive for the listed groups and parameters.
The paper cites [1] for the algorithm and provides raw results, but no proof or code for exhaustiveness is given.
domain assumption Difference-family equivalence and the additional automorphism-group/isomorphism filtering for order-4 unitals correctly separate isomorphism classes of designs.
The paper states DF equivalence is coarser than design isomorphism and that filtering sometimes does not catch isomorphic designs; no independent verification is supplied.
domain assumption GAP's SmallGroup library and LOOPS package generate faithful Cayley tables for the groups used.
Cayley tables are built with CayleyTable(IntoLoop(SmallGroup(v-1,idx))); the paper does not validate these tables independently.

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Cite this review

Pith. "Pith review of Enumerations of 1-rotational Steiner systems." pith.science (2026). https://pith.science/paper/6F6SARJP

@misc{pith2026250500469,
  author       = {Pith},
  title        = {Pith review of: Enumerations of 1-rotational Steiner systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6F6SARJP}},
  note         = {Machine review of arXiv:2505.00469}
}

abstract

In this paper new $1$-rotational 2-Steiner systems for different admissible $v,k$ pairs are introduced. In particular, $1$-rotational unitals of order $4$ are enumerated.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

7 extracted references · 5 canonical work pages

[4]

D´ avid Mez˝ oﬁ, G´ abor Nagy list of paramodiﬁed unitals https://davidmezoﬁ.github.io/unitals/

work page
[1]

Steiner systems S(2,6,121/126), S(2,7,169) based on difference families

I. Hetman Steiner systems S(2,6,121/126), S(2,7,169) based on diﬀerence families , 2024-2025. https://doi.org/10.48550/arXiv.2401.08274

work page Pith review arXiv doi:10.48550/arxiv.2401.08274 2024
[7]

1Rotational Steiner triple systems over arb itrary groups

Buratti, M. 1Rotational Steiner triple systems over arb itrary groups. Journal of Combinatorial Designs - J COMB DES. 9. 215-226. (2001) https://doi.org/10.1002/jcd .1008

doi:10.1002/jcd 2001
[3]

Krcadinac personal website https://web.math.pmf.u nizg.hr/ krcko/results/steiner.html

V. Krcadinac personal website https://web.math.pmf.u nizg.hr/ krcko/results/steiner.html

work page
[2]

C. J. Coulborn, J. H. Dinitz Handbook of Combinatorial Designs, Second Edition , 2007

work page 2007
[5]

The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.14.0 ; 2024, https://www.gap-system.org

2024
[6]

Nagy, G. P. and Vojtˇ echovsk´ y, P., loops, Computing with quasigroups and loops in GAP, Version 3.4.4 (2024) (GAP package), https://gap-packages.github.io/loops/

work page 2024

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [4]

D´ avid Mez˝ oﬁ, G´ abor Nagy list of paramodiﬁed unitals https://davidmezoﬁ.github.io/unitals/

work page

[2] [1]

Steiner systems S(2,6,121/126), S(2,7,169) based on difference families

I. Hetman Steiner systems S(2,6,121/126), S(2,7,169) based on diﬀerence families , 2024-2025. https://doi.org/10.48550/arXiv.2401.08274

work page Pith review arXiv doi:10.48550/arxiv.2401.08274 2024

[3] [7]

1Rotational Steiner triple systems over arb itrary groups

Buratti, M. 1Rotational Steiner triple systems over arb itrary groups. Journal of Combinatorial Designs - J COMB DES. 9. 215-226. (2001) https://doi.org/10.1002/jcd .1008

doi:10.1002/jcd 2001

[4] [3]

Krcadinac personal website https://web.math.pmf.u nizg.hr/ krcko/results/steiner.html

V. Krcadinac personal website https://web.math.pmf.u nizg.hr/ krcko/results/steiner.html

work page

[5] [2]

C. J. Coulborn, J. H. Dinitz Handbook of Combinatorial Designs, Second Edition , 2007

work page 2007

[6] [5]

The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.14.0 ; 2024, https://www.gap-system.org

2024

[7] [6]

Nagy, G. P. and Vojtˇ echovsk´ y, P., loops, Computing with quasigroups and loops in GAP, Version 3.4.4 (2024) (GAP package), https://gap-packages.github.io/loops/

work page 2024