# Algebraic characterization of regular fractions under level permutations

## TL;DR

This paper investigates how permutations of factor levels affect the regularity of fractions in factorial designs, introducing algebraic methods to identify when a symmetric orthogonal array can be transformed into a regular fraction.

## Contribution

It presents novel algebraic techniques using complex coding and polynomial algebra to determine the transformability of orthogonal arrays into regular fractions.

## Key findings

- Methods successfully identify when symmetric orthogonal arrays can be transformed into regular fractions.
- Techniques are demonstrated with examples involving factors with five levels.
- The approach provides a systematic way to analyze level permutations in factorial designs.

## Abstract

In this paper we study the behavior of the fractions of a factorial design under permutations of the factor levels. We focus on the notion of regular fraction and we introduce methods to check whether a given symmetric orthogonal array can or can not be transformed into a regular fraction by means of suitable permutations of the factor levels. The proposed techniques take advantage of the complex coding of the factor levels and of some tools from polynomial algebra. Several examples are described, mainly involving factors with five levels.

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Source: https://tomesphere.com/paper/1705.01340