Gray Code

A Gray code is an encoding of numbers so that adjacent numbers have a single digit differing by 1. The term Gray code is often used to refer to a "reflected" code, or more specifically still, the binary reflected Gray code.

To convert a binary number d_1d_2...d_(n-1)d_n to its corresponding binary reflected Gray code, start at the right with the digit d_n (the nth, or last, digit). If the d_(n-1) is 1, replace d_n by 1-d_n; otherwise, leave it unchanged. Then proceed to d_(n-1). Continue up to the first digit d_1, which is kept the same since d_0 is assumed to be a 0. The resulting number g_1g_2...g_(n-1)g_n is the reflected binary Gray code.

To convert a binary reflected Gray code g_1g_2...g_(n-1)g_n to a binary number, start again with the nth digit, and compute

 Sigma_n=sum_(i=1)^(n-1)g_i (mod 2).

If Sigma_n is 1, replace g_n by 1-g_n; otherwise, leave it the unchanged. Next compute

 Sigma_(n-1)=sum_(i=1)^(n-2)g_i (mod 2),

and so on. The resulting number d_1d_2...d_(n-1)d_n is the binary number corresponding to the initial binary reflected Gray code.

The code is called reflected because it can be generated in the following manner. Take the Gray code 0, 1. Write it forwards, then backwards: 0, 1, 1, 0. Then prepend 0s to the first half and 1s to the second half: 00, 01, 11, 10. Continuing, write 00, 01, 11, 10, 10, 11, 01, 00 to obtain: 000, 001, 011, 010, 110, 111, 101, 100, ... (OEIS A014550). Each iteration therefore doubles the number of codes.

Binary plot of the Gray code

The plots above show the binary representation of the first 255 (top figure) and first 511 (bottom figure) Gray codes. The Gray codes corresponding to the first few nonnegative integers are given in the following table.


The binary reflected Gray code is closely related to the solutions of the tower of Hanoi and baguenaudier, as well as to Hamiltonian cycles of hypercube graphs (including direction reversals; Skiena 1990, p. 149).

See also

Baguenaudier, Binary, Hilbert Curve, Ryser Formula, Thue-Morse Sequence, Tower of Hanoi

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Gardner, M. "The Binary Gray Code." Ch. 2 in Knotted Doughnuts and Other Mathematical Entertainments. New York: W. H. Freeman, 1986.Gilbert, E. N. "Gray Codes and Paths on the n-Cube." Bell System Tech. J. 37, 815-826, 1958.Gray, F. "Pulse Code Communication." United States Patent Number 2632058. March 17, 1953.Nijenhuis, A. and Wilf, H. Combinatorial Algorithms for Computers and Calculators, 2nd ed. New York: Academic Press, 1978.Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. "Gray Codes." §20.2 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 886-888, 1992.Skiena, S. "Gray Code." §1.5.3 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 42-43 and 149, 1990.Sloane, N. J. A. Sequence A014550 in "The On-Line Encyclopedia of Integer Sequences."Vardi, I. Computational Recreations in Mathematica. Redwood City, CA: Addison-Wesley, pp. 111-112 and 246, 1991.Wilf, H. S. Combinatorial Algorithms: An Update. Philadelphia, PA: SIAM, 1989.

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Gray Code

Cite this as:

Weisstein, Eric W. "Gray Code." From MathWorld--A Wolfram Web Resource.

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