blocksdesign (version 3.7)

MOLS: Prime power MOLS from finite fields

Description

Constructs sets of mutually orthogonal Latin squares (MOLS) of dimension p**q where p is prime. The number of squares r can be any number less than p**q and the exponent q can be any value from 1 up to a maximum dependent on the choice of p, viz:

q < 13 for p = 2; q < 8 for p = 3; q < 6 for p = 5; q < 5 for p = 7; q < 4 for p = 11, 13, 17, 19; q < 3 for any other p < 100; q = 1 for any other p > 100.

Usage

MOLS(p, q, r)

Arguments

p

is any odd prime greater than one

q

is any suitable integer power (see description for available powers)

r

is any number of squares up to a maximum of p**q - 1

Value

Data frame of factor levels for rows, columns and treatmnent sets

Details

Generates MOLS by cyclic permuation of a basic Latin square L0 constructed from a vector of ordered elements X of a prime-power finite field of size p**q and a unit vector 1 of length p**q (see Chapter 1 of Raghavarao 1971).

\(L0 = crossprod(X,t(1)) + crosssprod(1,t(X))\)

The primitive polynomials for the MOLS generated by this package were extracted from the Table of Primitive Polynomials given in the Supplement to Hansen and Mullen (1992).

The output is a single data frame for a \(p**q x p**q\) square classified by rows and columns with a separate column for the allocation of treatments to each individual square.

References

Hansen, T. & Mullen, G. L. (1992) Primitive polynomials over finite fields, Mathematics of Computation, 59, 639-643 and Supplement.

Raghavarao D. (1971) Constructions and Combinatorial Problems in Design of Experimnents, Dover Publications, Inc.

Examples

# NOT RUN {
MOLS(2,3,7)
MOLS(3,2,4)
# }
# NOT RUN {
MOLS(3,3,4)
# }
# NOT RUN {
MOLS(23,2,2)
# }
# NOT RUN {
 
# }