The minimal LFSR also has period 1, so it's always within half-a-bit
of synchronized, so it's not a counterexample :)
OK, here's a better counterexample:
max [15 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 15]
sub [15 -1 -1 -1 -5 -1 -1 7 7 -1 -1 -5 -1 -1 -1 15]
It's fixed point; divide by 15 to get the -1..1 range we've been
discussing.
Note the application of MXOR to form a shift register with taps.
-Dave
:: :: ::
from Numeric import *
iota = arange
rho = reshape
def l(*es): return array(es)
maximal = rho(l(0,0,0,1,
1,0,0,1,
0,1,0,0,
0,0,1,0), (4,4))
submax = rho(l(0,0,0,1,
1,0,0,1,
0,1,0,1,
0,0,1,0), (4,4))
def cycle(m):
def mxor(m,v): return matrixmultiply(m,v) % 2
n = m.shape[0]
v = zeros(n) + 1
rs = []
for i in iota(2**n-1):
rs.append(v[0])
v = mxor(m,v)
return rs
def autocorr(seq):
def phi(v,n): return concatenate((v[n:],v[:n]))
def corr(a,b): return sum((a==b)-(a!=b))
return array([corr(seq,phi(seq,i))
for i in iota(len(seq)+1)])
print "max", autocorr(cycle(maximal))
print "sub", autocorr(cycle(submax))
