Paleytype Hadamard matrices
Description
Computes a Hadamard matrix of dimension (p+1)*2^k, where p is a prime,
and p+1 is a multiple of 4, using the Paley construction. Used by hadamard
.
Usage
1 2 3 
Arguments
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? 
Details
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)*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.
Value
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.
References
Cameron PJ (2005) Hadamard Matrices. http://designtheory.org/library/encyc/topics/had.pdf. In: The Encyclopedia of Design Theory http://designtheory.org/library/encyc/
See Also
hadamard
Examples
1 2 3 4 5 6 7  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)
