There exist Steiner systems $S(2,7,505)$, $S(2,7,589)$, and $S(2,8,624)$

Ivan Hetman

arxiv: 2604.04975 · v3 · submitted 2026-04-04 · 🧮 math.CO

There exist Steiner systems S(2,7,505), S(2,7,589), and S(2,8,624)

Ivan Hetman This is my paper

Pith reviewed 2026-05-14 21:00 UTC · model grok-4.3

classification 🧮 math.CO

keywords Steiner systemsblock designsS(2,7,v)S(2,8,v)existencecombinatorial designshandbook of combinatorial designs

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The pith

Steiner systems S(2,7,505), S(2,7,589) and S(2,8,624) exist by explicit construction.

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

The paper gives concrete collections of blocks that form two Steiner systems S(2,7,505), two S(2,7,589), and ten S(2,8,624). A Steiner system S(2,k,v) is a set of k-element blocks on v points such that every pair of points lies in exactly one block. These examples settle two previously undecided cases among the 21 open parameters for block size 7 and one among the 37 open parameters for block size 8 that were listed in the Handbook of Combinatorial Designs. A reader cares because the constructions turn open existence questions into settled positive results.

Core claim

Explicit collections of blocks are given that form two Steiner systems S(2,7,505), two Steiner systems S(2,7,589), and ten Steiner systems S(2,8,624). These constructions demonstrate that designs with the stated parameters exist and resolve the undecided cases noted in the Handbook of Combinatorial Designs.

What carries the argument

The listed collections of 7-subsets (respectively 8-subsets) on the given number of points that cover every pair exactly once.

Load-bearing premise

The presented collections of blocks actually form Steiner systems in which every pair of points lies in exactly one block.

What would settle it

A direct enumeration check on the listed blocks that confirms every pair appears in precisely one block and no block is repeated.

read the original abstract

In this note two Steiner systems $S(2,7,505)$, two Steiner systems $S(2,7,589)$, and ten Steiner systems $S(2,8,624)$ are presented. This resolves two of $21$ undecided cases for block designs with block length $7$, and one of $37$ cases for block designs with block length $8$, mentioned in Handbook of Combinatorial Designs.

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.

Desk Editor's Note private letter to a colleague

Hetman gives explicit constructions for three previously undecided Steiner system parameters from the handbook, which is useful if the blocks hold up under check.

read the letter

The main takeaway is that this note supplies explicit block collections for two S(2,7,505), two S(2,7,589), and ten S(2,8,624). Those exact parameter sets were listed as open in the Handbook of Combinatorial Designs, so the existence claims are new and fill specific gaps in the tables for block lengths 7 and 8. The paper does well by actually presenting the designs rather than just arguing abstractly about existence. In design theory, concrete constructions are what count for settling undecided cases, and the author has produced multiple examples for each set. The soft spot is verification. The text presents the blocks but gives no computational check, no code, and no certificate that every pair lies in exactly one block. For v=624 that means confirming the property across roughly 6942 blocks, which is feasible by machine but not shown or referenced here. A referee or reader would need to run their own enumeration or accept the author's word. This work is for specialists who track Steiner systems S(2,k,v) and maintain the handbook tables. Someone in that narrow area gets direct value from the new parameter resolutions, even if they verify the lists separately. I would send it to peer review. The claims are concrete enough to be tested, the parameters matter inside the subfield, and referees can ask for verification data or confirm the constructions themselves. It is not a broad theoretical advance, but it is a solid incremental step that deserves referee time.

Referee Report

1 major / 0 minor

Summary. The paper presents explicit constructions for two Steiner systems S(2,7,505), two Steiner systems S(2,7,589), and ten Steiner systems S(2,8,624). These resolve two of the 21 undecided cases for block size 7 and one of the 37 undecided cases for block size 8 listed in the Handbook of Combinatorial Designs.

Significance. If the presented block collections satisfy the Steiner property, the results would constitute a concrete advance in the existence theory of Steiner systems by supplying explicit examples for previously open parameter sets. The explicit nature of the constructions is a strength, as it makes the claims in principle falsifiable by direct verification.

major comments (1)

The manuscript supplies the block collections (or generation rules) for the claimed designs, yet reports no verification that every pair of points lies in exactly one block. For the S(2,8,624) systems (b ≈ 6942 blocks), exhaustive pair enumeration is computationally feasible and must be performed and documented to support the central existence claim; absence of this check leaves the soundness of the constructions unconfirmed.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive summary and for the constructive major comment. We agree that explicit verification strengthens the paper and will incorporate it in the revision.

read point-by-point responses

Referee: The manuscript supplies the block collections (or generation rules) for the claimed designs, yet reports no verification that every pair of points lies in exactly one block. For the S(2,8,624) systems (b ≈ 6942 blocks), exhaustive pair enumeration is computationally feasible and must be performed and documented to support the central existence claim; absence of this check leaves the soundness of the constructions unconfirmed.

Authors: We agree that the manuscript does not report an explicit verification step and that this should be added for completeness. In the revised manuscript we will include a dedicated subsection describing the verification procedure. For all designs we have run exhaustive pair-enumeration checks (using a standard O(b · k²) algorithm that counts the number of blocks containing each pair) and confirmed that every pair appears in exactly one block. For the S(2,8,624) systems the check is feasible on ordinary hardware and has been completed; the same holds for the smaller S(2,7,505) and S(2,7,589) systems. We will document the method, the implementation outline, and the confirmation that no over- or under-covered pairs were found. revision: yes

Circularity Check

0 steps flagged

No circularity: explicit constructions of Steiner systems

full rationale

The paper asserts existence of the listed Steiner systems solely by presenting their block collections (or generation rules). No derivation chain, equations, or parameters are involved that could reduce by construction to fitted inputs, self-definitions, or self-citation loops. The central claim is a direct constructive existence statement whose validity is independent of any internal renaming or ansatz smuggling; verification of the λ=1 property is an external computational check, not a circular reduction within the argument itself.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper adds no free parameters or invented entities. It relies only on the standard definition of a Steiner system S(2,k,v).

axioms (1)

standard math A Steiner system S(2,k,v) is a pair (V,B) with |V|=v and every 2-subset of V contained in exactly one block from B, where each block has size k.
This is the foundational definition invoked by the abstract.

pith-pipeline@v0.9.0 · 5367 in / 1102 out tokens · 37711 ms · 2026-05-14T21:00:04.310024+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

3 extracted references · 3 canonical work pages

[1]

Edited by Charles J

Handbook of Combinatorial Designs . Edited by Charles J. Colbourn and Jeffrey H. Dinitz. Second edition. Discrete Mathematics and its Applications (Boca Raton). Chapman & Hall/CRC, Boca Raton, FL, 2007

work page 2007
[2]

Hetman, There Exist Steiner Systems S(2,8,225) and S(2,9,289) , Journal of Combinatorial Designs (2026)

I. Hetman, There Exist Steiner Systems S(2,8,225) and S(2,9,289) , Journal of Combinatorial Designs (2026)

work page 2026
[3]

B. D. McKay, A. Piperino, Practical Graph Isomorphism, II , Journal of Symbolic Computation, 60, 94--112 (2014)

work page 2014

[1] [1]

Edited by Charles J

Handbook of Combinatorial Designs . Edited by Charles J. Colbourn and Jeffrey H. Dinitz. Second edition. Discrete Mathematics and its Applications (Boca Raton). Chapman & Hall/CRC, Boca Raton, FL, 2007

work page 2007

[2] [2]

Hetman, There Exist Steiner Systems S(2,8,225) and S(2,9,289) , Journal of Combinatorial Designs (2026)

I. Hetman, There Exist Steiner Systems S(2,8,225) and S(2,9,289) , Journal of Combinatorial Designs (2026)

work page 2026

[3] [3]

B. D. McKay, A. Piperino, Practical Graph Isomorphism, II , Journal of Symbolic Computation, 60, 94--112 (2014)

work page 2014