Gray codes

This is based on work done about 20 years ago with a former student Jim McShea.

Gray codes were introduced by Bell Labs physicist Frank Gray in the 1950s. As introduced, a Gray code is an ordering of the n-tuples in GF(2)^n = \{0,1\}^n such that two successive terms differ in only one position. A Gray code can be regarded as a Hamiltonian path in the cube graph. For example:

[[0, 0, 0], [1, 0, 0], [1, 1, 0], [0, 1, 0], [0, 1, 1], [1, 1, 1], [1, 0, 1], [0, 0, 1]]

These can be generalized to n-tuples of integers (mod m) in the obvious way.

Gray codes have several applications:

  • solving puzzles such as the Tower of Hanoi and the Brain [G],
  • analog-digital-converters (goniometers) [S],
  • Hamiltonian circuits in hypercubes [Gil] and Cayley graphs of Coxeter groups [CSW],
  • capanology (the study of bell-ringing) [W],
  • continuous space-filling curves [Gi],
  • classification of Venn diagrams [R],
  • design of communication codes,
  • and more (see Wikipedia).
brain-puzzle

The Brain puzzle

Here's a SageMath/Python function for computing Gray codes.
def graycode(length,modulus):
    """
    Returns the n-tuple reverse Gray code mod m.


    EXAMPLES:
        sage: graycode(2,4)
        
        [[0, 0],
         [1, 0],
         [2, 0],
         [3, 0],
         [3, 1],
         [2, 1],
         [1, 1],
         [0, 1],
         [0, 2],
         [1, 2],
         [2, 2],
         [3, 2],
         [3, 3],
         [2, 3],
         [1, 3],
         [0, 3]]

    """
    n,m = length,modulus
    F = range(m)
    if n == 1:
        return [[i] for i in F]
    L = graycode(n-1, m)
    M = []
    for j in F:
        M = M+[ll+[j] for ll in L]
    k = len(M)
    Mr = [0]*m
    for i in range(m-1):
        i1 = i*int(k/m)
        i2 = (i+1)*int(k/m)
        Mr[i] = M[i1:i2]
    Mr[m-1] = M[(m-1)*int(k/m):]
    for i in range(m):
        if is_odd(i):
            Mr[i].reverse()
    M0 = []
    for i in range(m):
        M0 = M0+Mr[i]
    return M0


REFERENCES

[CSW] J. Conway, N. Sloane, and A. Wilks, “Gray codes and reflection groups”, Graphs and combinatorics 5(1989)315-325

[E] M. C. Er, “On generating the N-ary reflected Gray codes”, IEEE transactions on computers, 33(1984)739-741

[G] M. Gardner, “The binary Gray code”, in Knotted donuts and other mathematical entertainments, F. H. Freeman and Co., NY, 1986

[Gi] W. Gilbert, “A cube-filling Hilbert curve”, Math Intell 6 (1984)78

[Gil] E. Gilbert, “Gray codes and paths on the n-cube”, Bell System Technical Journal 37 (1958)815-826

[R] F. Ruskey, “A Survey of Venn Diagrams“, Elec. J. of Comb.(1997), and updated versions.

[S] Web page of T. Sillke

[W] A. White, “Ringing the cosets”, Amer. Math. Monthly 94(1987)721-746

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