`hadamard.Rd`

Returns a Hadamard matrix of dimension larger than the argument.

`hadamard(n)`

- n
lower bound for size

A Hadamard matrix

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 \(n>100\).

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

is a Hadamard matrix

Sloane NJA. A Library of Hadamard Matrices http://neilsloane.com/hadamard/

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/

```
par(mfrow=c(2,2))
## Sylvester-type
image(hadamard(63),main=quote("Sylvester: "*64==2^6))
## Paley-type
image(hadamard(59),main=quote("Paley: "*60==59+1))
## from NJ Sloane's library
image(hadamard(27),main=quote("Stored: "*28))
## For n=90 we get 96 rather than the minimum possible size, 92.
image(hadamard(90),main=quote("Constructed: "*96==2^3%*%(11+1)))
par(mfrow=c(1,1))
plot(2:150,sapply(2:150,function(i) ncol(hadamard(i))),type="S",
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))
```