Nothing
GF <- function(p,n){
primen100 <- c(2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,
61,67,71,73,79,83,89,97)
if (!(p %in% primen100)){stop("p is not a prime number less than 100.")}
## Now obsolete: require(conf.design); if( (p%%1) || (p < 2) ){stop("p is not a prime number.")}; if( primes(p)[length(primes(p))]!=p ){stop("p is not a prime number.")}
if( (n%%1) || (n < 1) ){stop("n is not a positive integer.")}
ord <- p^n # order of the field
if( n==1 ){ # trivial case
elemente <- c(0:(p-1))
primpol <- NULL
out<-list(elemente,primpol)
} else {
a <- factor.comb(p,n) # The monic polynomials are constructed using a full factorial plan
g <- cbind( rep(1,ord), a )
## evaluate polynomials for 0,1,...,p-1
## rows where h is all non-zero correspond to irreducible polynomials
## get them into irr
h <- matrix(0, ord, p)
for (i in 0:(p-1)){
a <- numeric(ord)
for (j in 1:n){
a <- a + g[,j]*(i^(n+1-j))
}
h[,i+1] <- (a + g[,n+1])%%p
}
irr <- g[!apply(h==0, 1, any), , drop = FALSE]
## Get the primitive roots. For all polynomials compute : p^1,p^2,...,p^(ord-1).
## Cycle is completed after at most ord-1 powers. If ord-1 powers are needed, a primitive root is found.
## Look for smallest power of p that has coefficients equal to (0,...,0,0,1).
## First power is always equal to (0,...,0,1,0), don't need to check that.
nirr <- nrow(irr)
z1 <- numeric(nirr)
for (i in 1:nirr){
dummy <- c(numeric(ord-3),1,0) # 1st power of (1,0); ord-3>0 since p,n>2.
j <- 0
while( z1[i]==0 ){ # ord-1st to 2nd power of (1,0)
j <- j+1
dummy <- redu.modp(c(dummy,0), irr[i,], p)
## division modulo p
if( all(dummy==c(numeric(n-1),1)) ) {z1[i] <- j+1}
## dummy is a polynomial of degree n-1.
## all(...)==TRUE if the result by division modulo p is (0,1), i.e. the number one.
## Here rest(x^n : polynom) = rest(x*rest(x^n-1 : polynom))
## get smallest power that equals (0,1)
}
}
primpol <- irr[which(z1==(ord-1)),,drop=FALSE]
elemente <- matrix(0,ord,n,byrow=TRUE)
elemente[2,n] <- 1 # 1st element is =0,...,0; 2nd element is 0,...0,1;
elemente[3,n-1] <- 1 # 3rd element ist =0,..,0,1,0.
dummy <- c(numeric(ord-3),1,0) # 1st power of (1,0); ord-3>0 since p,n>2.
for( i in 4:ord ){ # 2nd ord-2nd power of (1,0)
dummy <- redu.modp(c(dummy,0), primpol[1,], p)
elemente[i,] <- dummy
}
out<-list(elemente,primpol)
}
out
}
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