## Abstract

Semi-Latin rectangles are generalizations of Latin squares and semi-Latin

squares. Although they are called rectangles, the number of rows and the

number of columns are not necessarily distinct. There are k treatments in

each cell (row-column intersection): these constitute a block. Each treatment

of the design appears a definite number of times in each row and also a definite

number of times in each column (these parameters also being not necessarily

distinct). When k = 2, the design is said to have block size two. Regular-

graph semi-Latin rectangles have the additional property that the treatment

concurrences between any two pairs of distinct treatments differ by at most

one. Constructions for semi-Latin rectangles of this class with k = 2 which

have v treatments, v/2 rows and v columns, where v is even, are given in

Bailey and Monod (2001). These give the smallest designs when v is even.

Here we give constructions for smallest designs with k = 2 when v is odd.

These are regular-graph semi-Latin rectangles where the numbers of rows,

columns and treatments are identical. Then we extend the smallest designs

in each case to obtain larger designs.

squares. Although they are called rectangles, the number of rows and the

number of columns are not necessarily distinct. There are k treatments in

each cell (row-column intersection): these constitute a block. Each treatment

of the design appears a definite number of times in each row and also a definite

number of times in each column (these parameters also being not necessarily

distinct). When k = 2, the design is said to have block size two. Regular-

graph semi-Latin rectangles have the additional property that the treatment

concurrences between any two pairs of distinct treatments differ by at most

one. Constructions for semi-Latin rectangles of this class with k = 2 which

have v treatments, v/2 rows and v columns, where v is even, are given in

Bailey and Monod (2001). These give the smallest designs when v is even.

Here we give constructions for smallest designs with k = 2 when v is odd.

These are regular-graph semi-Latin rectangles where the numbers of rows,

columns and treatments are identical. Then we extend the smallest designs

in each case to obtain larger designs.

Original language | English |
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Pages (from-to) | 81-89 |

Journal | Journal of Statistical Planning and Inference |

Volume | 221 |

Early online date | 8 Mar 2022 |

DOIs | |

Publication status | Published - Dec 2022 |

## Keywords

- Regular-graph semi-Latin rectangle
- Starter
- Bi-starter
- Balanced tournament design
- Trojan square
- Undirected terrace