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arxiv: 2512.08681 · v2 · submitted 2025-12-09 · 🧮 math.CO · cs.DM

Recognition: 2 theorem links

· Lean Theorem

Resolvable Triple Arrays

Alexey Gordeev , Lars-Daniel \"Ohman
Authors on Pith no claims yet

Pith reviewed 2026-05-16 23:54 UTC · model grok-4.3

classification 🧮 math.CO cs.DM
keywords triple arraysresolvable designssymmetric 2-designsaffine planescombinatorial constructionsunordered triple arrays
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The pith

A new construction merges symmetric 2-designs with resolutions of other 2-designs to generate non-extremal triple arrays.

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

The paper presents a general method that produces triple arrays by taking a symmetric 2-design and combining it with a resolution of a second 2-design. This yields the first systematic supply of non-extremal examples, including new (21 × 15, 63)-arrays and a complete enumeration of the resolvable (7 × 15, 35) case. An infinite family of Paley triple arrays is shown to be resolvable, and every ((q + 1) × q², q(q + 1))-triple array is proved resolvable and in one-to-one correspondence with affine planes of order q. The authors also introduce unordered triple arrays as an intermediate object and offer a strengthened form of Agrawal’s conjecture on extremal cases.

Core claim

Resolvable triple arrays are defined as those obtainable by combining a symmetric 2-design with a resolution of another 2-design; this construction supplies the first general method for non-extremal triple arrays, produces concrete new examples, and establishes that all ((q + 1) × q², q(q + 1))-triple arrays are resolvable and correspond exactly to finite affine planes of order q.

What carries the argument

The resolvable triple-array construction obtained by combining a symmetric 2-design with a resolution of a second 2-design.

Load-bearing premise

The combined incidence structure from the two input designs automatically satisfies every triple-array condition once the designs exist for the target parameters.

What would settle it

A concrete ((q + 1) × q², q(q + 1))-triple array whose blocks cannot be partitioned to match the parallel classes of any affine plane of order q.

Figures

Figures reproduced from arXiv: 2512.08681 by Alexey Gordeev, Lars-Daniel \"Ohman.

Figure 1
Figure 1. Figure 1: A (4 × 9, 12)-triple array and a representation of its underlying unordered triple array. One of the main problems in the study of triple arrays is the construction of a triple array on a given set of admissible parameters. Since any triple array T has an underlying unordered triple array UT , one way to attack this problem is to split it into two separate sequential problems: Problem 3.2 (UTA construction… view at source ↗
Figure 2
Figure 2. Figure 2: A (3 × 4, 6)-unordered triple array. Note that there are no (3 × 4, 6)-triple arrays. For an (r × c, v)-unordered triple array U with row-sets R1, . . . , Rr and columns-sets C1, . . . , Cc, the UTA ordering problem can be viewed as a special case of the following problem, by setting Aij := Ri∩Cj . 5 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The (7 × 15, 35)-triple array from [20, 31] and a representation of its underlying unordered triple array. Construction 5.1 (RUTA construction). Let (r × c, v) be a parameter set admissible for triple arrays such that, in addition to e, λrc, λrr, λcc, the following two parameters are integers: λrrc := e(e − 1) r − 1 ∈ Z, k := c e ∈ Z. (4) Start with a symmetric 2-(r, r, e, e, λrrc) design S on the point se… view at source ↗
Figure 4
Figure 4. Figure 4: A (3 × 4, 6) and a (4 × 9, 12) resolvable unordered triple arrays. Example 5.6. Not all (unordered) triple arrays are resolvable, even when the corresponding λrrc and k are integers. Indeed, for the parameter set (7 × 8, 14), we have λrrc = 2 and k = 2, and two examples of (7 × 8, 14)-triple arrays, one resolvable and one not, are given in [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Two examples of (7 × 8, 14)-triple arrays and representations of their underlying unordered triple arrays. The top triple array is resolvable, while the bottom one is not. Although the transpose of an (unordered) triple array is always an (unordered) triple array, the next lemma shows that it can only be resolvable with regards to one orientation. 9 [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 3
Figure 3. Figure 3: A more detailed investigation of this case is presented in Section 7.4, where we enumerate and [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 6
Figure 6. Figure 6: Two examples of non-isotopic (5 × 6, 10)-triple arrays sharing the same underlying unordered triple array. We can therefore define an equivalence relation on triple arrays based on whether they have the same underlying unordered triple array. For some further context on this relation, an intercalate in a triple array is a 2 × 2 subsquare on two symbols, and flipping an intercalate means exchanging position… view at source ↗
read the original abstract

We present a new construction of triple arrays by combining a symmetric 2-design with a resolution of another 2-design. This is the first general method capable of producing non-extremal triple arrays. We call the triple arrays which can be obtained in this way resolvable. We employ the construction to produce the first examples of $(21 \times 15, 63)$-triple arrays, and enumerate all resolvable $(7 \times 15, 35)$-triple arrays, of which there was previously only a single known example. An infinite subfamily of Paley triple arrays turns out to be resolvable. We also introduce a new intermediate object, unordered triple arrays, that are to triple arrays what symmetric 2-designs are to Youden rectangles, and propose a strengthening of Agrawal's long-standing conjecture on the existence of extremal triple arrays. For small parameters, we completely enumerate all unordered triple arrays, and use this data to corroborate the new conjecture. We construct several infinite families of resolvable unordered triple arrays, and, in particular, show that all $((q + 1) \times q^2, q(q + 1))$-triple arrays are resolvable and are in correspondence with finite affine planes of order $q$.

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 / 3 minor

Summary. The paper introduces resolvable triple arrays constructed by combining a symmetric 2-design with a resolution of another 2-design, claiming this yields the first general method for non-extremal triple arrays. It produces the first examples of (21 × 15, 63)-triple arrays, enumerates all resolvable (7 × 15, 35)-triple arrays, shows an infinite subfamily of Paley triple arrays are resolvable, introduces unordered triple arrays as an intermediate object analogous to symmetric 2-designs relative to Youden rectangles, proposes a strengthening of Agrawal's conjecture on extremal triple arrays, and proves that all ((q + 1) × q², q(q + 1))-triple arrays are resolvable and correspond to affine planes of order q, supported by complete enumerations for small parameters.

Significance. If the constructions and bijection hold, this advances combinatorial design theory by supplying the first systematic construction for non-extremal triple arrays and forging a direct link to affine planes, which may enable further classifications and constructions. The enumerations provide concrete data corroborating the strengthened conjecture, while the new intermediate object (unordered triple arrays) offers a useful conceptual bridge to existing design theory.

major comments (2)
  1. [Product construction] In the product construction section: the claim that combining a symmetric 2-design with a resolvable 2-design always produces a triple array whose incidence parameters match exactly requires an explicit, general computation of the three lambda values (pairwise and triple incidences) to confirm constancy without hidden restrictions on the input designs.
  2. [Affine plane correspondence] In the section establishing the correspondence: the assertion that every ((q + 1) × q², q(q + 1))-triple array is resolvable and bijects with affine planes of order q is central; the argument mapping resolution classes to parallel classes must be expanded to prove that the resolution is canonically determined by the triple-array incidence structure alone.
minor comments (3)
  1. [Abstract] Abstract: the term 'unordered triple arrays' appears without definition; a parenthetical gloss would improve accessibility for readers unfamiliar with the new intermediate object.
  2. [Enumerations] Enumeration results: the counts for small parameters (including q=2,3 matching known affine-plane numbers) are stated in text; a compact table summarizing the numbers of resolvable and unordered triple arrays per parameter set would make the data easier to reference.
  3. [Paley family] Paley subfamily: the infinite family of resolvable Paley triple arrays is asserted; specifying the precise arithmetic condition on q (e.g., q ≡ 3 mod 4) under which the subfamily is resolvable would clarify the scope.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading, positive assessment, and constructive comments on our manuscript. We address each major comment below and will incorporate clarifications in the revised version.

read point-by-point responses

  1. Referee: [Product construction] In the product construction section: the claim that combining a symmetric 2-design with a resolvable 2-design always produces a triple array whose incidence parameters match exactly requires an explicit, general computation of the three lambda values (pairwise and triple incidences) to confirm constancy without hidden restrictions on the input designs.

    Authors: We thank the referee for this suggestion. While the construction is stated in full generality, we agree that an explicit computation of the incidence parameters would strengthen the presentation. In the revised manuscript we will insert a dedicated lemma providing the general formulas for the three lambda values (pairwise and triple incidences) and verify that they are constant for any symmetric 2-design and any resolvable 2-design, with no additional restrictions required on the input parameters. revision: yes

  2. Referee: [Affine plane correspondence] In the section establishing the correspondence: the assertion that every ((q + 1) × q², q(q + 1))-triple array is resolvable and bijects with affine planes of order q is central; the argument mapping resolution classes to parallel classes must be expanded to prove that the resolution is canonically determined by the triple-array incidence structure alone.

    Authors: We agree that the bijection is a central result and that the argument should be expanded for clarity. In the revision we will enlarge the relevant section with a detailed proof that any resolution of such a triple array is uniquely determined by its incidence structure alone, by showing that the resolution classes must coincide with the parallel classes of the associated affine plane. This will make the canonical nature of the correspondence explicit. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivations rely on independent standard designs

full rationale

The paper defines resolvable triple arrays via an explicit product construction from a symmetric 2-design and a resolvable 2-design, with direct incidence calculations verifying the lambda parameters. The claimed correspondence between ((q+1)×q², q(q+1))-triple arrays and affine planes of order q is realized by an explicit mapping of parallel classes to resolution classes, confirmed by matching small-q enumerations to known affine-plane counts. All steps use externally defined combinatorial objects and direct verification rather than self-referential definitions, fitted parameters renamed as predictions, or load-bearing self-citations. The enumeration of unordered triple arrays and the strengthened conjecture are likewise grounded in explicit computation for small parameters and standard constructions for infinite families.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

The central claims rest on the standard axioms of 2-designs and resolutions together with the newly introduced definitions of resolvable and unordered triple arrays.

axioms (1)
  • standard math Existence and incidence properties of symmetric 2-designs and resolvable 2-designs
    These are background objects whose properties are invoked to define the new construction.
invented entities (2)
  • resolvable triple array no independent evidence
    purpose: Triple arrays obtained via the stated combination of designs
    New class defined by the construction method.
  • unordered triple array no independent evidence
    purpose: Intermediate object for enumeration and conjecture testing
    Newly introduced simplification analogous to symmetric designs versus Youden rectangles.

pith-pipeline@v0.9.0 · 5516 in / 1366 out tokens · 37153 ms · 2026-05-16T23:54:08.467053+00:00 · methodology

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Reference graph

Works this paper leans on

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