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# 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 to its corresponding binary reflected Gray code, start at the right with the digit (the th, or last, digit). If the is 1, replace by ; otherwise, leave it unchanged. Then proceed to . Continue up to the first digit , which is kept the same since is assumed to be a 0. The resulting number is the reflected binary Gray code.

To convert a binary reflected Gray code to a binary number, start again with the th digit, and compute

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

and so on. The resulting number 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.

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.

 0 0 20 11110 40 111100 1 1 21 11111 41 111101 2 11 22 11101 42 111111 3 10 23 11100 43 111110 4 110 24 10100 44 111010 5 111 25 10101 45 111011 6 101 26 10111 46 111001 7 100 27 10110 47 111000 8 1100 28 10010 48 101000 9 1101 29 10011 49 101001 10 1111 30 10001 50 101011 11 1110 31 10000 51 101010 12 1010 32 110000 52 101110 13 1011 33 110001 53 101111 14 1001 34 110011 54 101101 15 1000 35 110010 55 101100 16 11000 36 110110 56 100100 17 11001 37 110111 57 100101 18 11011 38 110101 58 100111 19 11010 39 110100 59 100110

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).

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

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## References

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 -Cube." Bell System Tech. J. 37, 815-826, 1958.Gray, F. "Pulse Code Communication." United States Patent Number . 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.

Gray Code

## Cite this as:

Weisstein, Eric W. "Gray Code." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GrayCode.html