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)
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.
Minimum size for matrix
Maximum size for matrix. Ignored if prime is specified.
Optional. A prime at least as large as
n, such that prime+1 is divisible by 4.
Check that the resulting matrix is of Hadamard type
Matrix
"0/1" for a matrix of 0s and 1s, "+-" for a
matrix of \(\pm 1\).
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.
Cameron PJ (2005) Hadamard Matrices. In: The Encyclopedia of Design Theory http://www.maths.qmul.ac.uk/~lsoicher/designtheory.org/library/encyc/
hadamard
M<-paley(11)
is.hadamard(M)
## internals of is.hadamard(M)
H<-2*M-1
## HH^T is diagonal for any Hadamard matrix
H%*%t(H)
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