Embedding cyclic latin squares of order 2n in a complete set of orthogonal F-squares

S. J. Schwager; W. T. Federer; J. P. Mandeli

doi:10.1016/0378-3758(84)90073-9

Embedding cyclic latin squares of order 2ⁿ in a complete set of orthogonal F-squares

S. J. Schwager, W. T. Federer, J. P. Mandeli

Research output: Contribution to journal › Article › peer-review

5 Scopus citations

Abstract

A cyclic Latin square of order 2ⁿ, which has no orthogonal Latin square mate, is shown to have (2ⁿ-1)(2ⁿ-2) mutually orthogonal F(2ⁿ;2^n-1,2^n-1)-squares. This is a complete set of F-squares for the cyclic Latin square. Row and column operations areused to construct this complete set of F-squares from a Hadamard matrix and 2ⁿ-1 OF(2ⁿ;2^n-1,2^n-1)-squares into which the Latin square is decomposed. Tables of complete sets of mutually orthogonal F(2ⁿ;2^n-1,2^n-1)-squares are given for n=2 and 3, i.e., for cyclic Latin squares of orders 4 and 8.

Original language	English
Pages (from-to)	207-218
Number of pages	12
Journal	Journal of Statistical Planning and Inference
Volume	10
Issue number	2
DOIs	https://doi.org/10.1016/0378-3758(84)90073-9
State	Published - Aug 1984
Externally published	Yes

Keywords

F-square design
Factorial design
Hadamard matrix
Hadamard product
Latin square
Orthogonal F-squares

Access to Document

10.1016/0378-3758(84)90073-9

Cite this

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title = "Embedding cyclic latin squares of order 2n in a complete set of orthogonal F-squares",

abstract = "A cyclic Latin square of order 2n, which has no orthogonal Latin square mate, is shown to have (2n-1)(2n-2) mutually orthogonal F(2n;2n-1,2n-1)-squares. This is a complete set of F-squares for the cyclic Latin square. Row and column operations areused to construct this complete set of F-squares from a Hadamard matrix and 2n-1 OF(2n;2n-1,2n-1)-squares into which the Latin square is decomposed. Tables of complete sets of mutually orthogonal F(2n;2n-1,2n-1)-squares are given for n=2 and 3, i.e., for cyclic Latin squares of orders 4 and 8.",

keywords = "F-square design, Factorial design, Hadamard matrix, Hadamard product, Latin square, Orthogonal F-squares",

author = "Schwager, {S. J.} and Federer, {W. T.} and Mandeli, {J. P.}",

year = "1984",

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T1 - Embedding cyclic latin squares of order 2n in a complete set of orthogonal F-squares

AU - Schwager, S. J.

AU - Federer, W. T.

AU - Mandeli, J. P.

PY - 1984/8

Y1 - 1984/8

N2 - A cyclic Latin square of order 2n, which has no orthogonal Latin square mate, is shown to have (2n-1)(2n-2) mutually orthogonal F(2n;2n-1,2n-1)-squares. This is a complete set of F-squares for the cyclic Latin square. Row and column operations areused to construct this complete set of F-squares from a Hadamard matrix and 2n-1 OF(2n;2n-1,2n-1)-squares into which the Latin square is decomposed. Tables of complete sets of mutually orthogonal F(2n;2n-1,2n-1)-squares are given for n=2 and 3, i.e., for cyclic Latin squares of orders 4 and 8.

AB - A cyclic Latin square of order 2n, which has no orthogonal Latin square mate, is shown to have (2n-1)(2n-2) mutually orthogonal F(2n;2n-1,2n-1)-squares. This is a complete set of F-squares for the cyclic Latin square. Row and column operations areused to construct this complete set of F-squares from a Hadamard matrix and 2n-1 OF(2n;2n-1,2n-1)-squares into which the Latin square is decomposed. Tables of complete sets of mutually orthogonal F(2n;2n-1,2n-1)-squares are given for n=2 and 3, i.e., for cyclic Latin squares of orders 4 and 8.

KW - F-square design

KW - Factorial design

KW - Hadamard matrix

KW - Hadamard product

KW - Latin square

KW - Orthogonal F-squares

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