For a communication system, I need a special kind of gray codes. The requirements are:
One example of such Gray code is, for 5 bits and mrl = 4:
01111000011110000111100001111000
00111100000011111100001111110000
00011110000111100001111000011110
00001111110000111111000000111100
00000000111111110000000011111111
This paper give the best mrl values for different number of bits. Howerver, those values are found "By use of exhaustive computer searches"
I have python code that work well for small number of bits, up to 6:
N = 5 # number of bit
mrl = 4 # minimum run length
first_transition = [0]
first_code = [0]
def Recur(previous_transitions, previous_codes):
if len(previous_codes) == (2**N):
for b in xrange(N):
print ''.join([str((code >> (N-b-1)) & 1) for code in previous_codes])
print
return
new_transition_list = range(N)
for new_transition in new_transition_list:
ok = True
for i in xrange(mrl-1): #look back for transitions that are too close
try:
if new_transition == previous_transitions[-1-i]:
ok = False
break
except: break
if ok:
new_code = previous_codes[-1] ^ 2**new_transition #look back for repeated code
if not (new_code in previous_codes):
Recur(previous_transitions+[new_transition], previous_codes+[new_code])
Recur(first_transition, first_code )
raw_input('[end]')
My problem is that I would like a code of 20 bits, and the complexity of the basic approach seems close to O(n^3). Any suggestions on how to improve this code? Is there a better approach?
This is a (poor) python implementation of Method 1 described in Gray Codes with Optimized Run Lengths with special case for n=10 bits from Binary gray codes with long bit runs
I tried to use same terminology and variable names as in mentioned paper. I believe method 2 from the 1st paper might be able to improve some of the found gaps.
Let me know if useful, I can wrap it in a python package, or make a faster implementation in say rust.
import numpy as np
def transition_to_code( transition_sequence ):
code_sequence = [0]
n = np.int( np.log2( len(transition_sequence) ) )
code = 0
for pos in transition_sequence:
code ^= 1 << int(pos)
code_sequence.append(code)
return code_sequence[:-1]
def print_code_from_transition( transition_sequence ):
n = np.int( np.log2( len(transition_sequence) ) )
codes = transition_to_code( transition_sequence )
format_string = "b: {:0"+ str(n) +"b}"
for c in codes:
print( format_string.format( c ) )
def gap( transition_sequence ):
as_array = a = np.array( transition_sequence )
gap = 1
while gap < len(transition_sequence):
if np.any( as_array == np.roll(as_array, gap) ):
return gap
gap += 1
return 0
def valid_circuit( transition_sequence ):
n = np.int( np.log2( len(transition_sequence) ) )
if not len(transition_sequence) == 2**n:
print('Length not valid')
return False
if not np.all(np.array(transition_sequence) < n):
print('Transition codes not valid')
return False
sorted_codes = np.sort( transition_to_code( transition_sequence ) )
if not np.all( sorted_codes == np.arange(0,2**n) ):
print('Not all Unique')
return False
return True
transitions = {
2 : [0, 1, 0, 1],
3 : [0, 1, 0, 2, 0, 1, 0, 2],
4 : [0, 1, 2, 3, 2, 1, 0, 2, 0, 3, 0, 1, 3, 2, 3, 1],
5 : [0, 1, 2, 3, 4, 1, 2, 3, 0, 1, 4, 3, 2, 1, 4, 3, 0, 1, 2, 3, 4, 1, 2, 3, 0, 1, 4, 3, 2, 1, 4, 3],
6 : [0, 1, 2, 3, 4, 5, 0, 2, 4, 1, 3, 2, 0, 5, 4, 2, 3, 1, 4, 0, 2, 5, 3, 4, 2, 1, 0, 4, 3, 5, 2, 4, 0, 1, 2, 3, 4, 5, 0, 2, 4, 1, 3, 2, 0, 5, 4, 2, 3, 1, 4, 0, 2, 5, 3, 4, 2, 1, 0, 4, 3, 5, 2, 4]
}
def interleave(A, B):
n = np.int( np.log2( len(A) ) )
m = np.int( np.log2( len(B) ) )
M = 2**m
N = 2**n
assert N >= M
gap_A = gap(A)
gap_B = gap(B)
assert gap_A >= gap_B
st_pairs = [ (i, M-i) for i in range(M) if i % 2 == 1]
sorted_pairs = sorted(st_pairs, key=lambda p: np.abs(p[1]/p[0] - gap_B/gap_A) )
best_pair = sorted_pairs[0]
s, t = best_pair
ratio = t/s
P = "b"
while len(P) < M:
b_to_a_ratio = P.count('b') / (P.count('a') + 1)
if b_to_a_ratio >= ratio :
P += 'a'
else:
P += 'b'
return P * N
def P_to_transition(P, A, B):
Z = []
pos_a = 0
pos_b = 0
n = np.int( np.log2( len(A) ) )
delta = n
for p in P:
if p == 'a' :
Z.append( A[pos_a % len(A)] )
pos_a += 1
else :
Z.append( B[pos_b % len(B)] + delta )
pos_b += 1
return Z
"""
Code for special case for 10-bits to fabric a gap of 8.
From: Binary gray codes with long bit runs
by: Luis Goddyn∗ & Pavol Gvozdjak
"""
T0 = [0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3]
def to_4( i, sequence ):
permutations = []
indices = [j for j, x in enumerate(sequence) if x == i]
for pos in indices:
permutation = sequence.copy()
permutation[pos] = 4
permutations.append( permutation )
return permutations
def T_to_group(T):
state = np.array([0,0,0,0,0])
cycle = []
for pos in T:
cycle.append( state.copy() )
state[pos] += 1
state[pos] %= 4
return np.array( cycle )
def T_to_transition(T):
ticker = [False, False, False, False, False]
transitions = []
for t in T:
transistion = 2*t + 1*ticker[t]
ticker[t] = not ticker[t]
transitions.append( transistion )
return transitions
T1 = to_4( 0, T0)[3] * 4
T2 = to_4( 1, T1)[0] * 4
T3 = to_4( 2, T2)[1] * 4
transitions[10] = T_to_transition(T3)
for bits in range(2,21):
if bits in transitions:
print( "gray code for {} bits has gap: {}".format(bits, gap(transitions[bits]) ) )
else:
print( "finding code for {} bits...".format(bits) )
all_partitions = [ (i, bits-i) for i in range(bits) if i > 1]
partitions = [ (n, m) for (n,m) in all_partitions if n >= m and m > 1]
current_gap = 0
for n,m in partitions:
P = interleave( transitions[n], transitions[m])
Z = P_to_transition(P, transitions[n], transitions[m])
candidate_gap = gap( Z )
if candidate_gap > current_gap:
current_gap = candidate_gap
transitions[bits] = Z
if valid_circuit(transitions[bits]):
print( "gray code for {} bits has gap: {}".format(bits, gap(transitions[bits]) ) )
else:
print( "found in-valid gray code")
The loop above produces
gray code for 2 bits has gap: 2
gray code for 3 bits has gap: 2
gray code for 4 bits has gap: 2
gray code for 5 bits has gap: 4
gray code for 6 bits has gap: 4
finding code for 7 bits...
gray code for 7 bits has gap: 5
finding code for 8 bits...
gray code for 8 bits has gap: 5
finding code for 9 bits...
gray code for 9 bits has gap: 6
gray code for 10 bits has gap: 8
finding code for 11 bits...
gray code for 11 bits has gap: 8
finding code for 12 bits...
gray code for 12 bits has gap: 8
finding code for 13 bits...
gray code for 13 bits has gap: 9
finding code for 14 bits...
gray code for 14 bits has gap: 9
finding code for 15 bits...
gray code for 15 bits has gap: 11
finding code for 16 bits...
gray code for 16 bits has gap: 11
finding code for 17 bits...
gray code for 17 bits has gap: 12
finding code for 18 bits...
gray code for 18 bits has gap: 12
finding code for 19 bits...
gray code for 19 bits has gap: 13
finding code for 20 bits...
gray code for 20 bits has gap: 15
use transitions[3] or print_code_from_transition( transitions[3] ) to display the gray codes (in this example for 3 bits)