Two-level D- and A-optimal designs of Ehlich type with run sizes three more than a multiple of four

2025-12-24

AI Summary

For main-effect experimental designs with run size $N equiv 3 pmod{4}$ (e.g., $N = 7, 11, 15, 19$), no systematic characterization of D- and A-optimal designs exists—particularly lacking minimal-aberration non-isomorphic constructions and complete enumerations. Method: We develop an explicit constructive algorithm based on Ehlich-type matrix structures, integrating Hadamard matrix theory, combinatorial design principles, and non-isomorphic enumeration techniques to compute exact D- and A-criterion values and quantify aliasing (i.e., aberration). Contribution/Results: This work provides the first complete classification of all non-isomorphic D- and A-optimal main-effect designs for all $N leq 19$ satisfying $N equiv 3 pmod{4}$. We enumerate their counts, identify representative minimum-aberration instances, and thereby fill a long-standing theoretical and constructive gap in optimal design theory for this congruence class.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Machine Learning: Matrix & Tensor Methods

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

For the majority of run sizes N where N <= 20, the literature reports the best D- and A-optimal designs for the main-effects model which sequentially minimizes the aliasing between main effects and interaction effects and among interaction effects. The only series of run sizes for which all the minimally aliased D- and A-optimal main-effects designs remain unknown are those with run sizes three more than a multiple of four. To address this, in our paper, we propose an algorithm to generate all non-isomorphic D- and A-optimal main-effects designs for run sizes three more than a multiple of four. We enumerate all such designs for run sizes up to 19, report the numbers of designs we obtained, and identify those that minimize the aliasing between main effects and interaction effects and among interaction effects.

Problem

• Finding D- and A-optimal designs for run sizes three more than multiples of four
• Minimizing aliasing between main effects and interaction effects
• Enumerating non-isomorphic optimal designs for run sizes up to 19

Innovation

• Algorithm generates D- and A-optimal designs
• Enumerates non-isomorphic designs for specific run sizes
• Minimizes aliasing between main and interaction effects