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Row-column factorial designs with strength at least 2.

Row-column factorial designs with strength at least 2.

Authors :
Rahim, Fahim
Cavenagh, Nicholas J.
Source :
Linear Algebra & its Applications. Jun2023, Vol. 667, p44-70. 27p.
Publication Year :
2023

Abstract

The q k (full) factorial design with replication λ is the multi-set consisting of λ occurrences of each element of each q -ary vector of length k ; we denote this by λ × [ q ] k. An m × n row-column factorial design q k of strength t is an arrangement of the elements of λ × [ q ] k into an m × n array (which we say is of type I k (m , n , q , t)) such that for each row (column), the set of vectors therein are the rows of an orthogonal array of degree k , size n (respectively, m), q levels and strength t. Such arrays are used in experimental design. In this context, for a row-column factorial design of strength t , all subsets of interactions of size at most t can be estimated without confounding by the row and column blocking factors. In this manuscript we study row-column factorial designs with strength t ≥ 2. Our results for strength t = 2 are as follows. For any prime power q and assuming 2 ≤ M ≤ N , we show that there exists an array of type I k (q M , q N , q , 2) if and only if k ≤ M + N , k ≤ (q M − 1) / (q − 1) and (k , M , q) ≠ (3 , 2 , 2). We find necessary and sufficient conditions for the existence of I k (4 m , n , 2 , 2) for small parameters. We also show that I k + α (2 α b , 2 k , 2 , 2) exists whenever α ≥ 2 and 2 α + α + 1 ≤ k < 2 α b − α , assuming there exists a Hadamard matrix of order 4 b. For t = 3 we focus on the binary case. Assuming M ≤ N , there exists an array of type I k (2 M , 2 N , 2 , 3) if and only if M ≥ 5 , k ≤ M + N and k ≤ 2 M − 1. Most of our constructions use linear algebra, often in application to existing orthogonal arrays and Hadamard matrices. [ABSTRACT FROM AUTHOR]

Details

Language :
English
ISSN :
00243795
Volume :
667
Database :
Academic Search Index
Journal :
Linear Algebra & its Applications
Publication Type :
Academic Journal
Accession number :
162894057
Full Text :
https://doi.org/10.1016/j.laa.2023.02.018