def dft(seq, s):
Accounting.start_task("dft")
n = len(seq)
omega = cmath.exp((2 * cmath.pi * 1j) / n)
res = sum(seq[k] * omega ** (k + s) for k in range(n))
Accounting.finish_task("dft")
return res
def check_sequence_invariants(aa, bb):
Accounting.start_task('check_sequence_invariants')
#r = check_diophantine_invariant(aa, bb)
#if not r:
# Accounting.finish_task('check_sequence_invariants')
# return False
r = check_paf_invariant(aa, bb)
if not r:
Accounting.finish_task('check_sequence_invariants')
return False
#r = check_psd_invariant(aa, bb)
Accounting.finish_task('check_sequence_invariants')
return r
def check_diophantine_invariant(A, B):
Accounting.start_task('check_diophantine_invariant')
v = len(A)
a = sum(A)
b = sum(B)
# p.279 in "New Results..."
#
# By pre- and post-multiplying equation (1) with J_v, one obtains that
# a^2 + b^2 = 4v−2,
# where a and b are row sums of A and B
#
Accounting.finish_task('check_diophantine_invariant')
return equal(a ** 2 + b ** 2, 4*v - 2)
def check_psd_invariant(A, B):
Accounting.start_task('check_psd_invariant')
v = len(A)
# p.280 in "New Results..."
#
# define a circulant D-optimal matrix implies that the sum
# PSD_A(s) + PSD_B(s), s ≠ 0, is equal to the constant 2v − 2
#
cond = 2*v - 2
for k in range(1, v+1):
Accounting.start_task('check_psd_invariant_step')
# trick here:
# non negativity of PSD. we can discard before computing PSD both times
# but first I have to deal with rounding error. psd > cond with very
# low residual
psd_a = psd(A, k)
psd_b = psd(B, k)
if not equal(psd_a + psd_b, cond):
Accounting.finish_task('check_psd_invariant_step')
Accounting.finish_task('check_psd_invariant')
return False
Accounting.finish_task('check_psd_invariant_step')
Accounting.finish_task('check_psd_invariant')
return True
def check_paf_invariant(A, B):
# only check half of sequence due to PAF symmetry
Accounting.start_task('check_paf_invariant')
v = len(A)
for s in range(1, len(A)//2+1):
Accounting.start_task('check_paf_invariant_step')
paf_a = paf(A, s)
paf_b = paf(B, s)
if not paf_a + paf_b == 2:
Accounting.finish_task('check_paf_invariant_step')
Accounting.finish_task('check_paf_invariant')
return False
Accounting.finish_task('check_paf_invariant_step')
Accounting.finish_task('check_paf_invariant')
return True
def paf(seq, i):
Accounting.start_task("paf")
n = len(seq)
res = sum(seq[k] * seq[(k + i) % n] for k in range(n))
Accounting.finish_task("paf")
return res
def psd(seq, s):
Accounting.start_task("psd")
d = dft(seq, s)
magnitude = d.real ** 2 + d.imag ** 2
Accounting.finish_task("psd")
return magnitude
print("max_possible:", max_possible)
iter_mod = max_possible // 23
for aa in all_possible_sequences(N):
for bb in all_possible_sequences(N):
iterations += 1
percent_done = iterations / max_possible
if iterations % iter_mod == 0 or iterations == max_possible:
print("Percent checked: {0:>7.3f}% Matches found: {1}".format(
percent_done * 100, len(matches)
))
Accounting.start_task('check_sequence')
r = check_sequence_invariants(aa, bb)
if not r:
Accounting.finish_task('check_sequence')
continue
# all invariants hold!
matches.append((aa, bb))
Accounting.finish_task('check_sequence')
Accounting.finish_task('_program')
print("Done.\n")
print("Found {0} sequences out of {1} possible.".format(
len(matches), (2**N)**2)
)
for idx, match in enumerate(matches):
a, b = match
print("{:5}".format(idx+1), seq_to_str(a), seq_to_str(b))