# 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.

For `is.hadamard`, `TRUE` if `H` is a Hadamard matrix.

## References

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

`hadamard`
 ```1 2 3 4 5 6 7``` ```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) ```