Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Resolvable Triple Arrays

Alexey Gordeev, Lars-Daniel Öhman

TL;DR

This work provides the first general method to construct non-extremal (r × c, v)-triple arrays by combining a symmetric 2-design with a resolvable 2-design, yielding new resolvable examples such as (21 × 15, 63) and a complete enumeration of resolvable (7 × 15, 35) arrays. It introduces unordered triple arrays (UTAs) and develops a two-step approach (UTA construction and ordering) to tackle existence, revealing deep ties to finite geometry and affine planes. The paper also proves that all ((q+1) × q^2, q(q+1))-triple arrays are resolvable and correspond to affine planes of order q, provides infinite families of resolvable UTAs, and analyzes Paley-type constructions with respect to resolvability. Computational work with exact covers and automorphism analysis supports extensive enumeration in extremal and some non-extremal cases, while several open questions remain, especially around non-extremal parameter sets and broader UTAtoTA realizations.

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 $q$.

Resolvable Triple Arrays

TL;DR

This work provides the first general method to construct non-extremal (r × c, v)-triple arrays by combining a symmetric 2-design with a resolvable 2-design, yielding new resolvable examples such as (21 × 15, 63) and a complete enumeration of resolvable (7 × 15, 35) arrays. It introduces unordered triple arrays (UTAs) and develops a two-step approach (UTA construction and ordering) to tackle existence, revealing deep ties to finite geometry and affine planes. The paper also proves that all ((q+1) × q^2, q(q+1))-triple arrays are resolvable and correspond to affine planes of order q, provides infinite families of resolvable UTAs, and analyzes Paley-type constructions with respect to resolvability. Computational work with exact covers and automorphism analysis supports extensive enumeration in extremal and some non-extremal cases, while several open questions remain, especially around non-extremal parameter sets and broader UTAtoTA realizations.

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 -triple arrays, and enumerate all resolvable -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 -triple arrays are resolvable and are in correspondence with finite affine planes of order .

Paper Structure

This paper contains 26 sections, 19 theorems, 19 equations, 8 figures, 11 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. 2-designs
  4. Finite geometries
  5. Triple arrays and component designs
  6. Unordered triple arrays
  7. Agrawal's construction
  8. Triple arrays via resolutions
  9. Resolvable triple arrays
  10. Quad arrays
  11. Infinite families of resolvable unordered triple arrays
  12. Resolvable Paley triple arrays
  13. Enumeration and examples
  14. Notions of equivalence
  15. Computational methods
  16. ...and 11 more sections

Key Result

Theorem 1.1

Any $(r \times c, v)$-triple array with $v > \max(r, c)$ has $v \geq r + c - 1$.

Figures (8)

  • Figure 1: A $(4 \times 9, 12)$-triple array and a representation of its underlying unordered triple array.
  • Figure 2: A $(3 \times 4, 6)$-unordered triple array. Note that there are no $(3 \times 4, 6)$-triple arrays.
  • Figure 3: The $(7 \times 15, 35)$-triple array from mcsorleyDoubleArraysTriple2005ayucas_structure_7x15 and a representation of its underlying unordered triple array.
  • Figure 4: A $(3 \times 4, 6)$ and a $(4 \times 9, 12)$ resolvable unordered triple arrays.
  • Figure 5: Two examples of $(7 \times 8, 14)$-triple arrays and representations of their underlying unordered triple arrays. The top triple array is resolvable, while the bottom one is not.
  • ...and 3 more figures

Theorems & Definitions (53)

  • Theorem 1.1
  • Definition 2.1
  • Theorem 2.2: Fisher's inequality
  • Theorem 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 3.1
  • Proposition 4.2
  • proof
  • Example 4.3
  • ...and 43 more