# paley: Paley-type Hadamard matrices In survey: Analysis of Complex Survey Samples

## 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 paley(n, nmax = 2 * n, prime=NULL, check=!is.null(prime)) is.hadamard(H, style=c("0/1","+-"), full.orthogonal.balance=TRUE)

## Arguments

 n Minimum size for matrix nmax Maximum size for matrix. Ignored if prime is specified. prime Optional. A prime at least as large as n, such that prime+1 is divisible by 4. check Check that the resulting matrix is of Hadamard type H Matrix style "0/1" for a matrix of 0s and 1s, "+-" for a matrix of +/-1. 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.

## References

Cameron PJ (2005) Hadamard Matrices. In: The Encyclopedia of Design Theory http://www.maths.qmul.ac.uk/~lsoicher/designtheory.org/library/encyc/