## Tuesday, 31 January 2017

### Generating Gray codes with python

Gray codes are often used in positioning systems, see the wiki article for more details. I want to generate gray codes with python, and I'd like to generate all possible gray codes for a specific number of positions and number of bits. The maximum number of positions for a gray code is 2^numbits i.e. a 4 bit gray code can have at most 16 positions. It can also have fewer. If you want the gray code to be circular i.e. the first and last code have hamming distance 1, then there needs to be an even number of positions. Odd is fine if you don't care if it is circular.

There are of course algorithms for generating gray codes, but a constraint solver works really well, and also allows you to generate all possible gray codes for a specific number of positions. The python code below is for generating gray codes using python-constraint:

`from constraint import *from itertools import productnumbits = 4numpos = 16# this generates a list of all binary vectors of numbits bitsvecs = list(product(*(((1,0),)*numbits)))problem = Problem()# we will have numpos variables, i.e. one var for each code,# each var is an index into vecs, has 2^numbits possible valsproblem.addVariables(list(range(numpos)), list(range(len(vecs))))# vecs holds our binary vectors. # we want True if hamming distance between vecs[a] and vecs[b] is 1# False otherwise. This will be our constraint. def hammingDist1(a,b):    hamming = sum([vecs[a][i]^vecs[b][i] for i in range(numbits)])    if hamming == 1: return True    return False# add the important constraintsfor i in range(numpos):    j = (i+1)%numpos    problem.addConstraint(hammingDist1,[i,j])# make sure all the vectors are different    problem.addConstraint(AllDifferentConstraint())# print out the first solution we come acrosssol = problem.getSolution()for i,j in enumerate(sol):    print(vecs[sol[i]])    `

The main thing to note is that we add constraints so that each code has a hamming distance of 1 with its neighbour, and also that all the codes are diferent.

An example run looks like this (for a 4-bit, 16 position code):

`C:\Users\james\Desktop>python gray.py(0, 0, 0, 0)(0, 0, 0, 1)(0, 0, 1, 1)(0, 0, 1, 0)(0, 1, 1, 0)(0, 1, 1, 1)(0, 1, 0, 1)(1, 1, 0, 1)(1, 0, 0, 1)(1, 0, 0, 0)(1, 0, 1, 0)(1, 0, 1, 1)(1, 1, 1, 1)(1, 1, 1, 0)(1, 1, 0, 0)(0, 1, 0, 0)`

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