paley {survey}  R Documentation 
Computes a Hadamard matrix of dimension (p+1)\times 2^k
, where p is a prime,
and p+1 is a multiple of 4, using the Paley construction. Used by hadamard
.
paley(n, nmax = 2 * n, prime=NULL, check=!is.null(prime))
is.hadamard(H, style=c("0/1","+"), full.orthogonal.balance=TRUE)
n 
Minimum size for matrix 
nmax 
Maximum size for matrix. Ignored if 
prime 
Optional. A prime at least as large as

check 
Check that the resulting matrix is of Hadamard type 
H 
Matrix 
style 

full.orthogonal.balance 
Require full orthogonal balance? 
The Paley construction gives a Hadamard matrix of order p+1 if p is
prime and p+1 is a multiple of 4. This is then expanded to order
(p+1)\times 2^k
using the Sylvester construction.
paley
knows primes up to 7919. The user can specify a prime
with the prime
argument, in which case a matrix of order
p+1
is constructed.
If check=TRUE
the code uses is.hadamard
to check that
the resulting matrix really is of Hadamard type, in the same way as in
the example below. As this test takes n^3
time it is
preferable to just be sure that prime
really is prime.
A Hadamard matrix including a row of 1s gives BRR designs where the average of the replicates for a linear statistic is exactly the full sample estimate. This property is called full orthogonal balance.
For paley
, a matrix of zeros and ones, or NULL
if no matrix smaller than
nmax
can be found.
For is.hadamard
, TRUE
if H
is a Hadamard matrix.
Cameron PJ (2005) Hadamard Matrices. In: The Encyclopedia of Design Theory http://www.maths.qmul.ac.uk/~lsoicher/designtheory.org/library/encyc/
M<paley(11)
is.hadamard(M)
## internals of is.hadamard(M)
H<2*M1
## HH^T is diagonal for any Hadamard matrix
H%*%t(H)