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arxiv: 2406.19516 · v2 · submitted 2024-06-27 · 🧮 math.CO · cs.DM· math.OC

Almost Orthogonal Arrays: Search Three Ways

Luis Mart\'inez , Mar\'ia Merino , Juan Manuel Montoya , Josu\'e Tonelli-Cueto This is my paper

Pith reviewed 2026-05-23 23:53 UTC · model grok-4.3

classification 🧮 math.CO cs.DMmath.OC
keywords almost orthogonal arraysinteger programminglocal searchalgebraic constructionsnon-orthogonality measurescombinatorial designsexperimental design
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The pith

Three search methods produce almost orthogonal arrays that improve on existing constructions for multiple non-orthogonality measures.

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

Orthogonal arrays are difficult to build exactly for many parameter choices, so the paper studies their relaxation to almost orthogonal arrays that permit controlled departures from perfect orthogonality. The authors generate new examples by formulating the search as an integer program, by running local-search meta-heuristics, and by using algebraic constructions. When the resulting arrays are scored on several different non-orthogonality measures, many of them match or beat the best previously published arrays. All newly found arrays are placed in a public repository. The work therefore supplies a practical route to obtaining usable near-orthogonal designs without requiring exact orthogonality.

Core claim

Integer programming, local-search meta-heuristics, and algebraic methods each locate almost orthogonal arrays whose quality, measured by several non-orthogonality criteria, is competitive with or strictly better than the arrays already recorded in the literature; the complete collection of new arrays is deposited in a public repository.

What carries the argument

Almost orthogonal arrays, obtained by relaxing the pairwise balance conditions of orthogonal arrays and located through integer-programming models, local-search heuristics, and algebraic constructions.

If this is right

  • Researchers needing near-orthogonal designs now have additional explicit arrays for many parameter sets.
  • When one of the three search techniques fails to improve an existing array, the other two remain available.
  • The public repository supplies a growing baseline that future constructions can be compared against.
  • The same three techniques can be applied immediately to parameter combinations not yet examined.

Where Pith is reading between the lines

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

  • Hybrid algorithms that combine integer programming with local search or algebraic insight could locate still better arrays.
  • Direct benchmarking inside a specific application domain would test whether lower non-orthogonality scores translate into measurable gains in that domain.
  • Algebraic constructions may expose structural patterns that purely numerical search methods overlook.

Load-bearing premise

The chosen non-orthogonality measures are reliable indicators of how useful the arrays will be in downstream applications.

What would settle it

A concrete experimental-design task in which an array with a worse score on every non-orthogonality measure used in the paper nevertheless produces better statistical results than the improved arrays would falsify the practical significance of the reported gains.

read the original abstract

Orthogonal arrays play a fundamental role in many applications. However, constructing orthogonal arrays with the required parameters for an application usually is extremely difficult and, sometimes, even impossible. Hence there is an increasing need for a relaxation of orthogonal arrays to allow a wider flexibility. The latter has lead to various types of arrays under the name of ``nearly-orthogonal arrays'', and less often ``almost orthogonal arrays''. In this paper, we explore how to find almost orthogonal arrays three ways: using integer programming, local search meta-heuristics and algebraic methods. We compare all our search results with the ones existing in the literature, and we show that they are competitive, improving some of the existing arrays for many non-orthogonality measures. All our found almost orthogonal arrays are available at a public repository.

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 investigates constructions of almost orthogonal arrays via three distinct search paradigms: integer programming, local-search metaheuristics, and algebraic techniques. It reports that the resulting arrays are competitive with, and in several cases improve upon, previously tabulated examples when evaluated under multiple non-orthogonality measures; all constructed arrays are deposited in a public repository.

Significance. If the numerical comparisons hold, the work supplies immediately usable arrays for experimental-design settings where exact orthogonality is unattainable, together with a reproducible data release that supports downstream verification and reuse. The multi-method strategy itself is a modest methodological contribution.

major comments (2)
  1. [§4, Tables 2–5] §4 (Results), Tables 2–5: the improvement claims rest on direct numerical comparison of non-orthogonality measures between newly constructed arrays and literature references. The manuscript does not supply the exact evaluation scripts or the precise floating-point implementations used for the reference arrays; an undetected difference in measure definition or rounding would falsify the reported gains. Because the arrays themselves are public, this verification gap is fixable but currently load-bearing for the central empirical claim.
  2. [§3.1] §3.1 (Integer Programming formulation): the objective function and the encoding of the non-orthogonality penalties are stated at a high level; the precise linearization or big-M constants employed for each measure are not given. Without these details, independent reproduction of the IP solutions (and therefore of the claimed improvements) cannot be confirmed.
minor comments (3)
  1. The repository URL is mentioned only in the abstract; it should also appear in the main text (e.g., at the end of §4) with a permanent identifier or commit hash.
  2. Notation for the various non-orthogonality measures is introduced piecemeal; a single consolidated table or subsection collecting all definitions would improve readability.
  3. A few figure captions (e.g., Figure 3) omit the precise parameter tuple (v,k,λ,…) of the displayed array, forcing the reader to cross-reference the tables.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the recommendation of minor revision. The comments correctly identify two reproducibility gaps that we will close by expanding the manuscript and supplementing the public repository. We address each point below.

read point-by-point responses

  1. Referee: [§4, Tables 2–5] §4 (Results), Tables 2–5: the improvement claims rest on direct numerical comparison of non-orthogonality measures between newly constructed arrays and literature references. The manuscript does not supply the exact evaluation scripts or the precise floating-point implementations used for the reference arrays; an undetected difference in measure definition or rounding would falsify the reported gains. Because the arrays themselves are public, this verification gap is fixable but currently load-bearing for the central empirical claim.

    Authors: We agree that the absence of the evaluation scripts creates a verification gap. In the revised manuscript we will add an appendix containing the complete Python scripts used to compute every non-orthogonality measure, including the exact floating-point implementations, rounding conventions, and measure definitions employed for both our arrays and the literature references. These scripts, together with the already-public arrays, will be deposited in the repository so that every tabulated comparison can be reproduced exactly. revision: yes

  2. Referee: [§3.1] §3.1 (Integer Programming formulation): the objective function and the encoding of the non-orthogonality penalties are stated at a high level; the precise linearization or big-M constants employed for each measure are not given. Without these details, independent reproduction of the IP solutions (and therefore of the claimed improvements) cannot be confirmed.

    Authors: We accept that the current presentation of the IP model is insufficient for independent reproduction. The revised §3.1 will contain the fully linearized objective function, the explicit big-M constants chosen for each penalty term, the complete set of auxiliary variables and constraints, and the Gurobi parameter settings used to obtain the reported solutions. revision: yes

Circularity Check

0 steps flagged

No circularity: purely computational search with external comparisons

full rationale

The paper reports results from three independent search algorithms (integer programming, local search, algebraic methods) applied to construct almost orthogonal arrays, followed by direct numerical comparison against published literature examples using explicitly defined non-orthogonality measures. No derivation chain, fitted parameters renamed as predictions, self-referential definitions, or load-bearing self-citations appear; the central claims rest on the correctness of the search outputs and the reproducibility of the measure calculations, which are external to any internal reduction. This matches the default expectation for computational enumeration papers.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the standard combinatorial definitions of orthogonal arrays and non-orthogonality measures drawn from prior literature; no new free parameters, axioms, or invented entities are introduced.

axioms (1)
  • domain assumption Standard definitions of orthogonal arrays and almost orthogonal arrays from combinatorial design theory
    The paper presupposes these definitions to frame the search problem.

pith-pipeline@v0.9.0 · 5674 in / 1067 out tokens · 16075 ms · 2026-05-23T23:53:13.437153+00:00 · methodology

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