def hadamard_matrix_paleyII(n):
N = ZZ(n/2)
if is_prime(N-1) and (N-1)%4==1:
p = N-1
else:
raise ValueError, "The order %s is not covered by the Paley type II construction."%n
S = matrix(ZZ,[[H2(i,j,p) for i in range(N)] for j in range(N)])
return block_matrix([[S+1,S-1],[S-1,-S-1]])
def hadamard_matrix(n):
if not(n%4==0) and not(n==2):
raise ValueError, "The Hadamard matrix of order %s does not exist"%n
if n==2:
return matrix([[1,1],[1,-1]])
if is_even(n):
N = ZZ(n/2)
elif n==1:
return matrix([1])
if is_prime(N-1) and (N-1)%4==1:
return hadamard_matrix_paleyII(n)
elif n==4 or n%8==0:
had = hadamard_matrix(ZZ(n/2))
chad1 = matrix([list(r)+list(r) for r in had.rows()])
mhad = (-1)*had
R = len(had.rows())
chad2 = matrix([list(had.rows()[i])+list(mhad.rows()[i]) for i in range(R)])
return chad1.stack(chad2)
elif is_prime(N-1) and (N-1)%4==3:
return hadamard_matrix_paleyI(n)
else:
raise ValueError, "The Hadamard matrix of order %s is not yet implemented."%n
def hadamard_matrix_paleyI(n):
if is_prime(n-1) and (n-1)%4==3:
p = n-1
else:
raise ValueError, "The order %s is not covered by the Paley type I construction."%n
return matrix(ZZ,[[H1(i,j,p) for i in range(n)] for j in range(n)])