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Row-column factorial designs with strength at least 2.
Row-column factorial designs with strength at least 2.
- 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