pith. sign in

arxiv: 2511.05191 · v3 · pith:VAEESLRXnew · submitted 2025-11-07 · 🧮 math.CO

Steiner systems S(2,6,226) and S(2,6,441) exist

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

Pith reviewed 2026-05-21 19:57 UTC · model grok-4.3

classification 🧮 math.CO
keywords Steiner systemS(2,6,v)1-rotational designpoint-transitive designcomputer searchcombinatorial designblock designexistence question
0
0 comments X

The pith

Computer searches construct seven non-isomorphic 1-rotational Steiner systems S(2,6,226) and six point-transitive Steiner systems S(2,6,441).

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

The paper uses computer enumeration to locate Steiner systems S(2,6,226) and S(2,6,441). It reports seven distinct 1-rotational copies for v=226 and six point-transitive copies for v=441. These constructions settle two of the 29 existence questions that remained open for Steiner systems with these parameters. Readers care because such systems realize balanced incomplete block designs that appear in coding, geometry, and scheduling problems.

Core claim

Via computer search, seven non-isomorphic 1-rotational Steiner systems S(2,6,226) and six point-transitive Steiner systems S(2,6,441) were found, resolving two of 29 previously undecided cases for S(2,6,v).

What carries the argument

1-rotational and point-transitive Steiner systems S(2,6,v), block designs in which every pair of points lies in exactly one 6-element block, searched under symmetry constraints that reduce the search space for computer enumeration.

If this is right

  • The existence of an S(2,6,226) is settled in the affirmative.
  • The existence of an S(2,6,441) is settled in the affirmative.
  • At least seven non-isomorphic 1-rotational S(2,6,226) exist.
  • At least six point-transitive S(2,6,441) exist.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same symmetry-restricted search technique could be applied to the remaining undecided values of v for S(2,6,v).
  • The newly constructed systems supply explicit examples that can be checked for extra properties such as resolvability or automorphism group size.
  • These designs may serve as base cases for recursive constructions that produce larger Steiner systems with the same block size.

Load-bearing premise

The computer enumeration and isomorphism-testing procedures correctly identify valid Steiner systems and distinguish non-isomorphic copies without false positives or missed isomorphisms.

What would settle it

An independent computer enumeration that finds no valid systems for either parameter set or that counts a different number of non-isomorphic examples would falsify the reported existence and multiplicity.

read the original abstract

Via computer search, we found seven non-isomorphic $1$-rotational Steiner systems $S(2,6,226)$ and six point-transitive Steiner systems $S(2,6,441)$, resolving two of $29$ previously undecided cases for $S(2,6,v)$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims that computer searches under 1-rotational and point-transitive symmetry constraints have located seven non-isomorphic Steiner systems S(2,6,226) and six point-transitive Steiner systems S(2,6,441), thereby resolving two of the 29 previously open existence cases for S(2,6,v).

Significance. If the computational findings hold, the result is a concrete advance in the existence spectrum for Steiner systems with block size 6. It supplies new admissible orders and demonstrates that symmetry-reduced enumeration remains an effective tool for settling large open cases in design theory.

major comments (2)
  1. [§3] §3 (Enumeration for v=226): the text states that seven non-isomorphic 1-rotational systems were found but does not exhibit the base blocks or orbit generators. Without these explicit representatives, independent verification that every pair lies in exactly one block cannot be performed from the manuscript alone.
  2. [§4] §4 (Enumeration for v=441): similarly, the six point-transitive systems are asserted to exist, yet no explicit block orbits or automorphism generators are supplied, leaving the pair-coverage check dependent on re-running the unreleased search implementation.
minor comments (2)
  1. [Introduction] The count of 29 undecided cases is given without a direct citation to the source survey or table; adding the reference would improve traceability.
  2. [Figure 1] Figure 1 (search tree diagram) uses abbreviations for group actions that are defined only later in the text; moving the legend earlier would aid readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation and for identifying the need for explicit data to support independent verification. We address the major comments point by point below.

read point-by-point responses

  1. Referee: [§3] §3 (Enumeration for v=226): the text states that seven non-isomorphic 1-rotational systems were found but does not exhibit the base blocks or orbit generators. Without these explicit representatives, independent verification that every pair lies in exactly one block cannot be performed from the manuscript alone.

    Authors: We agree that explicit representatives are required for full verifiability from the manuscript. In the revised version we will add the base blocks and orbit generators for all seven 1-rotational Steiner systems S(2,6,226), placed in an appendix. This will permit direct, independent confirmation that every pair is covered exactly once. revision: yes

  2. Referee: [§4] §4 (Enumeration for v=441): similarly, the six point-transitive systems are asserted to exist, yet no explicit block orbits or automorphism generators are supplied, leaving the pair-coverage check dependent on re-running the unreleased search implementation.

    Authors: We accept this observation. The revised manuscript will include the explicit block orbits and automorphism generators for the six point-transitive Steiner systems S(2,6,441). We will also make the search code available to readers upon request, thereby removing any dependence on an unreleased implementation. revision: yes

Circularity Check

0 steps flagged

No circularity: direct computational existence claim

full rationale

The paper reports the existence of specific Steiner systems S(2,6,226) and S(2,6,441) discovered via computer enumeration under 1-rotational and point-transitive symmetry constraints. This constitutes a direct verification against the standard definition of a Steiner system (every pair in exactly one block), with no equations, fitted parameters, predictions derived from subsets of the same data, or self-citations that reduce the claim to prior results by construction. The derivation chain consists of describing the search methodology and stating the count of non-isomorphic examples found; none of the enumerated circularity patterns apply, and the result is self-contained as an independent computational finding.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the standard axiomatic definition of a Steiner system S(2,6,v) and on the correctness of the computational enumeration under the stated symmetry groups. No free parameters are fitted; no new entities are postulated.

axioms (1)
  • standard math A Steiner system S(2,6,v) is a set of 6-subsets (blocks) of a v-set such that every 2-subset is contained in exactly one block.
    Invoked in the opening definition and used as the verification criterion for the computer output.

pith-pipeline@v0.9.0 · 5573 in / 1255 out tokens · 52949 ms · 2026-05-21T19:57:19.292472+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

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

  1. Linear Geometry: flats, ranks, regularity, parallelity

    math.HO 2025-11 unverdicted novelty 2.0

    A survey of foundational concepts in Linear Geometry including flats, ranks, regularity, modularity, and parallelity all derived from flat hulls.

Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages · cited by 1 Pith paper

  1. [1]

    Colbourn, J.H

    C.J. Colbourn, J.H. Dinitz (eds.),Handbook of combinatorial designs, Discrete Mathematics and its Applications, Chapman & Hall/CRC, Boca Raton, FL, 2007. xxii+984 pp

    work page 2007

  2. [2]

    Hetman,Point-transitive Steiner systemsS(2,6,111/121/126),S(2,7,169/175), preprint, 2025

    I. Hetman,Point-transitive Steiner systemsS(2,6,111/121/126),S(2,7,169/175), preprint, 2025. (https://doi.org/10.48550/arXiv.2504.14931)

    work page doi:10.48550/arxiv.2504.14931 2025

  3. [3]

    Wilson,An existence theory for pairwise balanced designs, I

    R.M. Wilson,An existence theory for pairwise balanced designs, I. Composition theorems and morphisms, J. Combinatorial Theory Ser. A13(1972), 220–245

    work page 1972

  4. [4]

    Wilson,An existence theory for pairwise balanced designs

    R.M. Wilson,An existence theory for pairwise balanced designs. II. The structure of PBD-closed sets and the existence conjectures, J. Combinatorial Theory Ser. A13(1972), 246–273

    work page 1972

  5. [5]

    Wilson,An existence theory for pairwise balanced designs

    R.M. Wilson,An existence theory for pairwise balanced designs. III. Proof of the existence conjectures, J. Combinatorial Theory Ser. A18(1975), 71–79. T. Banakh: Ivan Franko National University of L viv (Ukraine), and Jan Kochanowski University in Kielce (Poland) I. Hetman: L viv (Ukraine) A. Ravsky: Pidstryhach Institute for Applied Problems of Mechanics...

    work page 1975