### Description

Returns a Hadamard matrix of dimension larger than the argument.

### Usage

hadamard(n)


### Arguments

 n lower bound for size

### Details

For most n the matrix comes from paley. The 36\times 36 matrix is from Plackett and Burman (1946) and the 28\times 28 is from Sloane's library of Hadamard matrices.

Matrices of dimension every multiple of 4 are thought to exist, but this function doesn't know about all of them, so it will sometimes return matrices that are larger than necessary. The excess is at most 4 for n<180 and at most 5% for n>100.

### Note

Strictly speaking, a Hadamard matrix has entries +1 and -1 rather than 1 and 0, so 2*hadamard(n)-1 is a Hadamard matrix

### References

Plackett RL, Burman JP. (1946) The Design of Optimum Multifactorial Experiments Biometrika, Vol. 33, No. 4 pp. 305-325

Cameron PJ (2005) Hadamard Matrices http://designtheory.org/library/encyc/topics/had.pdf. In: The Encyclopedia of Design Theory http://designtheory.org/library/encyc/

brrweights, paley

### Examples


par(mfrow=c(2,2))
## Sylvester-type
## Paley-type
## from NJ Sloane's library
## For n=90 we get 96 rather than the minimum possible size, 92.

par(mfrow=c(1,1))
ylab="Matrix size",xlab="n",xlim=c(1,150),ylim=c(1,150))
abline(0,1,lty=3)
lines(2:150, 2:150-(2:150 %% 4)+4,col="purple",type="S",lty=2)
legend(c(x=10,y=140),legend=c("Actual size","Minimum possible size"),
col=c("black","purple"),bty="n",lty=c(1,2))



survey documentation built on May 3, 2023, 9:12 a.m.