Gray code

The reflected binary code (RBC), also known as reflected binary (RB) or Gray code after Frank Gray, is an ordering of the binary numeral system such that two successive values differ in only one bit (binary digit).

For example, the representation of the decimal value "1" in binary would normally be "001", and "2" would be "010". In Gray code, these values are represented as "001" and "011". That way, incrementing a value from 1 to 2 requires only one bit to change, instead of two.

Gray codes are widely used to prevent spurious output from electromechanical switches and to facilitate error correction in digital communications such as digital terrestrial television and some cable TV systems. The use of Gray code in these devices helps simplify logic operations and reduce errors in practice.

Function

Many devices indicate position by closing and opening switches. If that device uses natural binary codes, positions 3 and 4 are next to each other but all three bits of the binary representation differ:

Decimal	Binary
...	...
3	011
4	100
...	...

The problem with natural binary codes is that physical switches are not ideal: it is very unlikely that physical switches will change states exactly in synchrony. In the transition between the two states shown above, all three switches change state. In the brief period while all are changing, the switches will read some spurious position. Even without keybounce, the transition might look like 011 — 001 — 101 — 100. When the switches appear to be in position 001, the observer cannot tell if that is the "real" position 1, or a transitional state between two other positions. If the output feeds into a sequential system, possibly via combinational logic, then the sequential system may store a false value.

This problem can be solved by changing only one switch at a time, so there is never any ambiguity of position, resulting in codes assigning to each of a contiguous set of integers, or to each member of a circular list, a word of symbols such that no two code words are identical and each two adjacent code words differ by exactly one symbol. These codes are also known as unit-distance, single-distance, single-step, monostrophic or syncopic codes, in reference to the Hamming distance of 1 between adjacent codes.

Invention

File:Reflected binary Gray 2632058.png

Gray's patent introduces the term "reflected binary code"

In principle, there can be more than one such code for a given word length, but the term Gray code was first applied to a particular binary code for non-negative integers, the binary-reflected Gray code, or BRGC. Bell Labs researcher George R. Stibitz described such a code in a 1941 patent application, granted in 1943. Frank Gray introduced the term reflected binary code in his 1947 patent application, remarking that the code had "as yet no recognized name". He derived the name from the fact that it "may be built up from the conventional binary code by a sort of reflection process".

File:Gray code tesseract.svg

Visualized as a traversal of vertices of a tesseract

File:Gray code number line arcs.svg

Gray code along the number line

In the standard encoding of the Gray code the least significant bit follows a repetitive pattern of 2 on, 2 off (... 11001100 ...); the next digit a pattern of 4 on, 4 off; the i-th least significant bit a pattern of 2ⁱ on 2ⁱ off. The most significant digit is an exception to this: for an n-bit Gray code, the most significant digit follows the pattern 2ⁿ⁻¹ on, 2ⁿ⁻¹ off, which is the same (cyclic) sequence of values as for the second-most significant digit, but shifted forwards 2ⁿ⁻² places. The four-bit version of this is shown below:

Decimal	Binary	Gray
0	0000	0000
1	0001	0001
2	0010	0011
3	0011	0010
4	0100	0110
5	0101	0111
6	0110	0101
7	0111	0100
8	1000	1100
9	1001	1101
10	1010	1111
11	1011	1110
12	1100	1010
13	1101	1011
14	1110	1001
15	1111	1000

For decimal 15 the code rolls over to decimal 0 with only one switch change. This is called the cyclic or adjacency property of the code.

In modern digital communications, Gray codes play an important role in error correction. For example, in a digital modulation scheme such as QAM where data is typically transmitted in symbols of 4 bits or more, the signal's constellation diagram is arranged so that the bit patterns conveyed by adjacent constellation points differ by only one bit. By combining this with forward error correction capable of correcting single-bit errors, it is possible for a receiver to correct any transmission errors that cause a constellation point to deviate into the area of an adjacent point. This makes the transmission system less susceptible to noise.

Despite the fact that Stibitz described this code before Gray, the reflected binary code was later named after Gray by others who used it. Two different 1953 patent applications use "Gray code" as an alternative name for the "reflected binary code"; one of those also lists "minimum error code" and "cyclic permutation code" among the names. A 1954 patent application refers to "the Bell Telephone Gray code". Other names include "cyclic binary code", "cyclic progression code", "cyclic permuting binary" or "cyclic permuted binary" (CPB).

The Gray code is sometimes misattributed to 19th century electrical device inventor Elisha Gray.

History and practical application

Mathematical puzzles

Reflected binary codes were applied to mathematical puzzles before they became known to engineers.

The binary-reflected Gray code represents the underlying scheme of the classical Chinese rings puzzle, a sequential mechanical puzzle mechanism described by the French Louis Gros in 1872.

It can serve as a solution guide for the Towers of Hanoi problem, based on a game by the French Édouard Lucas in 1883. Similarly, the so-called Towers of Bucharest and Towers of Klagenfurt game configurations yield ternary and pentary Gray codes.

Martin Gardner wrote a popular account of the Gray code in his August 1972 "Mathematical Games" column in Scientific American.

The code also forms a Hamiltonian cycle on a hypercube, where each bit is seen as one dimension.

Telegraphy codes

When the French engineer Émile Baudot changed from using a 6-unit (6-bit) code to 5-unit code for his printing telegraph system, in 1875 or 1876, he ordered the alphabetic characters on his print wheel using a reflected binary code, and assigned the codes using only three of the bits to vowels. With vowels and consonants sorted in their alphabetical order, and other symbols appropriately placed, the 5-bit character code has been recognized as a reflected binary code. This code became known as Baudot code and, with minor changes, was eventually adopted as International Telegraph Alphabet No. 1 (ITA1, CCITT-1) in 1932.

About the same time, the German-Austrian Template:Ill demonstrated another printing telegraph in Vienna using a 5-bit reflected binary code for the same purpose, in 1874.

Analog-to-digital signal conversion

Frank Gray, who became famous for inventing the signaling method that came to be used for compatible color television, invented a method to convert analog signals to reflected binary code groups using vacuum tube-based apparatus. Filed in 1947, the method and apparatus were granted a patent in 1953, and the name of Gray stuck to the codes. The "PCM tube" apparatus that Gray patented was made by Raymond W. Sears of Bell Labs, working with Gray and William M. Goodall, who credited Gray for the idea of the reflected binary code.

File:US02632058 Gray.png

Part of front page of Gray's patent, showing PCM tube (10) with reflected binary code in plate (15)

Gray was most interested in using the codes to minimize errors in converting analog signals to digital; his codes are still used today for this purpose.

Position encoders

File:Encoder Disc (3-Bit).svg

Rotary encoder for angle-measuring devices marked in 3-bit binary-reflected Gray code (BRGC)

File:Gray code rotary encoder 13-track opened.jpg

A Gray code absolute rotary encoder with 13 tracks. Housing, interrupter disk, and light source are in the top; sensing element and support components are in the bottom.

Gray codes are used in linear and rotary position encoders (absolute encoders and quadrature encoders) in preference to weighted binary encoding. This avoids the possibility that, when multiple bits change in the binary representation of a position, a misread will result from some of the bits changing before others.

For example, some rotary encoders provide a disk which has an electrically conductive Gray code pattern on concentric rings (tracks). Each track has a stationary metal spring contact that provides electrical contact to the conductive code pattern. Together, these contacts produce output signals in the form of a Gray code. Other encoders employ non-contact mechanisms based on optical or magnetic sensors to produce the Gray code output signals.

Regardless of the mechanism or precision of a moving encoder, position measurement error can occur at specific positions (at code boundaries) because the code may be changing at the exact moment it is read (sampled). A binary output code could cause significant position measurement errors because it is impossible to make all bits change at exactly the same time. If, at the moment the position is sampled, some bits have changed and others have not, the sampled position will be incorrect. In the case of absolute encoders, the indicated position may be far away from the actual position and, in the case of incremental encoders, this can corrupt position tracking.

In contrast, the Gray code used by position encoders ensures that the codes for any two consecutive positions will differ by only one bit and, consequently, only one bit can change at a time. In this case, the maximum position error will be small, indicating a position adjacent to the actual position.

Genetic algorithms

Due to the Hamming distance properties of Gray codes, they are sometimes used in genetic algorithms. They are very useful in this field, since mutations in the code allow for mostly incremental changes, but occasionally a single bit-change can cause a big leap and lead to new properties.

Boolean circuit minimization

Gray codes are also used in labelling the axes of Karnaugh maps since 1953 as well as in Händler circle graphs since 1958, both graphical methods for logic circuit minimization.

Error correction

In modern digital communications, 1D- and 2D-Gray codes play an important role in error prevention before applying an error correction. For example, in a digital modulation scheme such as QAM where data is typically transmitted in symbols of 4 bits or more, the signal's constellation diagram is arranged so that the bit patterns conveyed by adjacent constellation points differ by only one bit. By combining this with forward error correction capable of correcting single-bit errors, it is possible for a receiver to correct any transmission errors that cause a constellation point to deviate into the area of an adjacent point. This makes the transmission system less susceptible to noise.

• Codes 4-PSK
  Codes 4-PSK
• Codes 8-PSK
  Codes 8-PSK
• Codes 16-QAM
  Codes 16-QAM

Communication between clock domains

Digital logic designers use Gray codes extensively for passing multi-bit count information between synchronous logic that operates at different clock frequencies. The logic is considered operating in different "clock domains". It is fundamental to the design of large chips that operate with many different clocking frequencies.

Cycling through states with minimal effort

If a system has to cycle sequentially through all possible combinations of on-off states of some set of controls, and the changes of the controls require non-trivial expense (e.g. time, wear, human work), a Gray code minimizes the number of setting changes to just one change for each combination of states. An example would be testing a piping system for all combinations of settings of its manually operated valves.

A balanced Gray code can be constructed, that flips every bit equally often. Since bit-flips are evenly distributed, this is optimal in the following way: balanced Gray codes minimize the maximal count of bit-flips for each digit.

Gray code counters and arithmetic

George R. Stibitz utilized a reflected binary code in a binary pulse counting device in 1941 already.

A typical use of Gray code counters is building a FIFO (first-in, first-out) data buffer that has read and write ports that exist in different clock domains. The input and output counters inside such a dual-port FIFO are often stored using Gray code to prevent invalid transient states from being captured when the count crosses clock domains. The updated read and write pointers need to be passed between clock domains when they change, to be able to track FIFO empty and full status in each domain. Each bit of the pointers is sampled non-deterministically for this clock domain transfer. So for each bit, either the old value or the new value is propagated. Therefore, if more than one bit in the multi-bit pointer is changing at the sampling point, a "wrong" binary value (neither new nor old) can be propagated. By guaranteeing only one bit can be changing, Gray codes guarantee that the only possible sampled values are the new or old multi-bit value. Typically Gray codes of power-of-two length are used.

Sometimes digital buses in electronic systems are used to convey quantities that can only increase or decrease by one at a time, for example the output of an event counter which is being passed between clock domains or to a digital-to-analog converter. The advantage of Gray codes in these applications is that differences in the propagation delays of the many wires that represent the bits of the code cannot cause the received value to go through states that are out of the Gray code sequence. This is similar to the advantage of Gray codes in the construction of mechanical encoders, however the source of the Gray code is an electronic counter in this case. The counter itself must count in Gray code, or if the counter runs in binary then the output value from the counter must be reclocked after it has been converted to Gray code, because when a value is converted from binary to Gray code, it is possible that differences in the arrival times of the binary data bits into the binary-to-Gray conversion circuit will mean that the code could go briefly through states that are wildly out of sequence. Adding a clocked register after the circuit that converts the count value to Gray code may introduce a clock cycle of latency, so counting directly in Gray code may be advantageous.

To produce the next count value in a Gray-code counter, it is necessary to have some combinational logic that will increment the current count value that is stored. One way to increment a Gray code number is to convert it into ordinary binary code, add one to it with a standard binary adder, and then convert the result back to Gray code. Other methods of counting in Gray code are discussed in a report by Robert W. Doran, including taking the output from the first latches of the master-slave flip flops in a binary ripple counter.

Gray code addressing

As the execution of executable code typically causes an instruction memory access pattern of locally consecutive addresses, bus encodings using Gray code addressing instead of binary addressing can reduce the number of state changes of the address bits significantly, thereby reducing the CPU power consumption in some low-power designs.

Constructing an n-bit Gray code

File:Binary-reflected Gray code construction.svg

The first few steps of the reflect-and-prefix method.

File:Gray code permutation matrix 16.svg

4-bit Gray code permutation

The binary-reflected Gray code list for n bits can be generated recursively from the list for n − 1 bits by reflecting the list (i.e. listing the entries in reverse order), prefixing the entries in the original list with a binary 0, prefixing the entries in the reflected list with a binary 1, and then concatenating the original list with the reversed list. For example, generating the n = 3 list from the n = 2 list:

2-bit list:	00, 01, 11, 10
Reflected:		10, 11, 01, 00
Prefix old entries with 0:	000, 001, 011, 010,
Prefix new entries with 1:		110, 111, 101, 100
Concatenated:	000, 001, 011, 010,	110, 111, 101, 100

The one-bit Gray code is G₁ = (0,1). This can be thought of as built recursively as above from a zero-bit Gray code G₀ = ( Λ ) consisting of a single entry of zero length. This iterative process of generating G_n+1 from G_n makes the following properties of the standard reflecting code clear:

• G_n is a permutation of the numbers 0, ..., 2ⁿ − 1. (Each number appears exactly once in the list.)
• G_n is embedded as the first half of G_n+1.
• Therefore, the coding is stable, in the sense that once a binary number appears in G_n it appears in the same position in all longer lists; so it makes sense to talk about the reflective Gray code value of a number: G(m) = the mth reflecting Gray code, counting from 0.
• Each entry in G_n differs by only one bit from the previous entry. (The Hamming distance is 1.)
• The last entry in G_n differs by only one bit from the first entry. (The code is cyclic.)

These characteristics suggest a simple and fast method of translating a binary value into the corresponding Gray code. Each bit is inverted if the next higher bit of the input value is set to one. This can be performed in parallel by a bit-shift and exclusive-or operation if they are available: the nth Gray code is obtained by computing $n\oplus \lfloor \frac{n}{2}\rfloor$. Prepending a 0 bit leaves the order of the code words unchanged, prepending a 1 bit reverses the order of the code words. If the bits at position $i$ of codewords are inverted, the order of neighbouring blocks of $2^i$ codewords is reversed. For example, if bit 0 is inverted in a 3 bit codeword sequence, the order of two neighbouring codewords is reversed

If bit 1 is inverted, blocks of 2 codewords change order: Template:Block indent

If bit 2 is inverted, blocks of 4 codewords reverse order: Template:Block indent

Thus, performing an exclusive or on a bit $b_i$ at position $i$ with the bit $b_{i+1}$ at position $i+1$ leaves the order of codewords intact if $b_{i+1}=\mathtt{0}$, and reverses the order of blocks of $2^{i+1}$ codewords if $b_{i+1}=\mathtt{1}$. Now, this is exactly the same operation as the reflect-and-prefix method to generate the Gray code.

A similar method can be used to perform the reverse translation, but the computation of each bit depends on the computed value of the next higher bit so it cannot be performed in parallel. Assuming $g_i$ is the $i$th Gray-coded bit ($g_0$ being the most significant bit), and $b_i$ is the $i$th binary-coded bit ($b_0$ being the most-significant bit), the reverse translation can be given recursively: $b_0=g_0$, and $b_i=g_i\oplus b_{i-1}$. Alternatively, decoding a Gray code into a binary number can be described as a prefix sum of the bits in the Gray code, where each individual summation operation in the prefix sum is performed modulo two.

To construct the binary-reflected Gray code iteratively, at step 0 start with the ${\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{e}}_0=\mathtt{0}$, and at step $i>0$ find the bit position of the least significant 1 in the binary representation of $i$ and flip the bit at that position in the previous code ${\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{e}}_{i-1}$ to get the next code ${\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{e}}_i$. The bit positions start 0, 1, 0, 2, 0, 1, 0, 3, ... See find first set for efficient algorithms to compute these values.

Converting to and from Gray code

The following functions in C convert between binary numbers and their associated Gray codes. While it may seem that Gray-to-binary conversion requires each bit to be handled one at a time, faster algorithms exist.

typedef unsigned int uint;

// This function converts an unsigned binary number to reflected binary Gray code.
uint BinaryToGray(uint num)
{
    return num ^ (num >> 1); // The operator >> is shift right. The operator ^ is exclusive or.
}

// This function converts a reflected binary Gray code number to a binary number.
uint GrayToBinary(uint num)
{
    uint mask = num;
    while (mask) {           // Each Gray code bit is exclusive-ored with all more significant bits.
        mask >>= 1;
        num   ^= mask;
    }
    return num;
}

// A more efficient version for Gray codes 32 bits or fewer through the use of SWAR (SIMD within a register) techniques. 
// It implements a parallel prefix XOR function. The assignment statements can be in any order.
// 
// This function can be adapted for longer Gray codes by adding steps.

uint GrayToBinary32(uint num)
{
    num ^= num >> 16;
    num ^= num >>  8;
    num ^= num >>  4;
    num ^= num >>  2;
    num ^= num >>  1;
    return num;
}
// A Four-bit-at-once variant changes a binary number (abcd)2 to (abcd)2 ^ (00ab)2, then to (abcd)2 ^ (00ab)2 ^ (0abc)2 ^ (000a)2.

On newer processors, the number of ALU instructions in the decoding step can be reduced by taking advantage of the CLMUL instruction set. If MASK is the constant binary string of ones ended with a single zero digit, then carryless multiplication of MASK with the grey encoding of x will always give either x or its bitwise negation.

Special types of Gray codes

In practice, "Gray code" almost always refers to a binary-reflected Gray code (BRGC). However, mathematicians have discovered other kinds of Gray codes. Like BRGCs, each consists of a list of words, where each word differs from the next in only one digit (each word has a Hamming distance of 1 from the next word).

Gray codes with n bits and of length less than 2ⁿ

It is possible to construct binary Gray codes with n bits with a length of less than 2ⁿ, if the length is even. One possibility is to start with a balanced Gray code and remove pairs of values at either the beginning and the end, or in the middle. OEIS sequence A290772 gives the number of possible Gray sequences of length 2n that include zero and use the minimum number of bits.

n-ary Gray code

Ternary number → ternary Gray code Template:Monodiv

There are many specialized types of Gray codes other than the binary-reflected Gray code. One such type of Gray code is the n-ary Gray code, also known as a non-Boolean Gray code. As the name implies, this type of Gray code uses non-Boolean values in its encodings.

For example, a 3-ary (ternary) Gray code would use the values 0,1,2. The (n, k)-Gray code is the n-ary Gray code with k digits. The sequence of elements in the (3, 2)-Gray code is: 00,01,02,12,11,10,20,21,22. The (n, k)-Gray code may be constructed recursively, as the BRGC, or may be constructed iteratively. An algorithm to iteratively generate the (N, k)-Gray code is presented (in C):

// inputs: base, digits, value
// output: Gray
// Convert a value to a Gray code with the given base and digits.
// Iterating through a sequence of values would result in a sequence
// of Gray codes in which only one digit changes at a time.
void toGray(unsigned base, unsigned digits, unsigned value, unsigned gray[digits])
{ 
	unsigned baseN[digits];	// Stores the ordinary base-N number, one digit per entry
	unsigned i;		// The loop variable
 
	// Put the normal baseN number into the baseN array. For base 10, 109 
	// would be stored as [9,0,1]
	for (i = 0; i < digits; i++) {
		baseN[i] = value % base;
		value    = value / base;
	}
 
	// Convert the normal baseN number into the Gray code equivalent. Note that
	// the loop starts at the most significant digit and goes down.
	unsigned shift = 0;
	while (i--) {
		// The Gray digit gets shifted down by the sum of the higher
		// digits.
		gray[i] = (baseN[i] + shift) % base;
		shift = shift + base - gray[i];	// Subtract from base so shift is positive
	}
}
// EXAMPLES
// input: value = 1899, base = 10, digits = 4
// output: baseN[] = [9,9,8,1], gray[] = [0,1,7,1]
// input: value = 1900, base = 10, digits = 4
// output: baseN[] = [0,0,9,1], gray[] = [0,1,8,1]

There are other Gray code algorithms for (n,k)-Gray codes. The (n,k)-Gray code produced by the above algorithm is always cyclical; some algorithms, such as that by Guan, lack this property when k is odd. On the other hand, while only one digit at a time changes with this method, it can change by wrapping (looping from n − 1 to 0). In Guan's algorithm, the count alternately rises and falls, so that the numeric difference between two Gray code digits is always one.

Gray codes are not uniquely defined, because a permutation of the columns of such a code is a Gray code too. The above procedure produces a code in which the lower the significance of a digit, the more often it changes, making it similar to normal counting methods.

See also Skew binary number system, a variant ternary number system where at most two digits change on each increment, as each increment can be done with at most one digit carry operation.

Balanced Gray code

Although the binary reflected Gray code is useful in many scenarios, it is not optimal in certain cases because of a lack of "uniformity". In balanced Gray codes, the number of changes in different coordinate positions are as close as possible. To make this more precise, let G be an R-ary complete Gray cycle having transition sequence $({\delta }_k)$; the transition counts (spectrum) of G are the collection of integers defined by

$${\lambda }_k=|\{j\in {\mathbb{\mathbb{Z}}}_{R^n}:{\delta }_j=k\}|,\text{ for }k\in {\mathbb{\mathbb{Z}}}_n$$

A Gray code is uniform or uniformly balanced if its transition counts are all equal, in which case we have ${\lambda }_k=\frac{R^n}{n}$ for all k. Clearly, when $R=2$, such codes exist only if n is a power of 2. If n is not a power of 2, it is possible to construct well-balanced binary codes where the difference between two transition counts is at most 2; so that (combining both cases) every transition count is either $2\lfloor \frac{2^n}{2n}\rfloor$ or $2\lceil \frac{2^n}{2n}\rceil$. Gray codes can also be exponentially balanced if all of their transition counts are adjacent powers of two, and such codes exist for every power of two.

For example, a balanced 4-bit Gray code has 16 transitions, which can be evenly distributed among all four positions (four transitions per position), making it uniformly balanced: Template:Block indent Template:Block indent Template:Block indent Template:Block indent whereas a balanced 5-bit Gray code has a total of 32 transitions, which cannot be evenly distributed among the positions. In this example, four positions have six transitions each, and one has eight: Template:Block indent Template:Block indent Template:Block indent Template:Block indent Template:Block indent

We will now show a construction and implementation for well-balanced binary Gray codes which allows us to generate an n-digit balanced Gray code for every n. The main principle is to inductively construct an (n + 2)-digit Gray code $G^{\prime }$ given an n-digit Gray code G in such a way that the balanced property is preserved. To do this, we consider partitions of $G=g_0,\dots ,g_{2^n-1}$ into an even number L of non-empty blocks of the form

$$\left\{g_0\right\},\{g_1,\dots ,g_{k_2}\},\{g_{k_2+1},\dots ,g_{k_3}\},\dots ,\{g_{k_{L-2}+1},\dots ,g_{-2}\},\left\{g_{-1}\right\}$$

where $k_1=0$, $k_{L-1}=-2$, and $k_L\equiv -1(\mathrm{mod}{2}^n)$). This partition induces an $(n+2)$-digit Gray code given by

If we define the transition multiplicities

$$m_i=\left|\{j:{\delta }_{k_j}=i,1\le j\le L\}\right|$$

to be the number of times the digit in position i changes between consecutive blocks in a partition, then for the (n + 2)-digit Gray code induced by this partition the transition spectrum $\lambda {\text{'}}_i$ is

$$\lambda {\text{'}}_i=\{\begin{array}{ll}4{\lambda }_i-2m_i,& \text{if }0\le i<n\\ L,& \text{ otherwise }\end{array}$$

The delicate part of this construction is to find an adequate partitioning of a balanced n-digit Gray code such that the code induced by it remains balanced, but for this only the transition multiplicities matter; joining two consecutive blocks over a digit $i$ transition and splitting another block at another digit $i$ transition produces a different Gray code with exactly the same transition spectrum $\lambda {\text{'}}_i$, so one may for example designate the first $m_i$ transitions at digit $i$ as those that fall between two blocks. Uniform codes can be found when $R\equiv 0(\mathrm{mod}4)$ and $R^n\equiv 0(\mathrm{mod}n)$, and this construction can be extended to the R-ary case as well.

Long run Gray codes

Long run (or maximum gap) Gray codes maximize the distance between consecutive changes of digits in the same position. That is, the minimum run-length of any bit remains unchanged for as long as possible.

Monotonic Gray codes

Monotonic codes are useful in the theory of interconnection networks, especially for minimizing dilation for linear arrays of processors. If we define the weight of a binary string to be the number of 1s in the string, then although we clearly cannot have a Gray code with strictly increasing weight, we may want to approximate this by having the code run through two adjacent weights before reaching the next one.

We can formalize the concept of monotone Gray codes as follows: consider the partition of the hypercube $Q_n=(V_n,E_n)$ into levels of vertices that have equal weight, i.e.

$$V_n(i)=\{v\in V_n:v\text{ has weight }i\}$$

for $0\le i\le n$. These levels satisfy $|V_n(i)|=(\genfrac{}{}{0ex}{}{n}{i})$. Let $Q_n(i)$ be the subgraph of $Q_n$ induced by $V_n(i)\cup V_n(i+1)$, and let $E_n(i)$ be the edges in $Q_n(i)$. A monotonic Gray code is then a Hamiltonian path in $Q_n$ such that whenever ${\delta }_1\in E_n(i)$ comes before ${\delta }_2\in E_n(j)$ in the path, then $i\le j$.

An elegant construction of monotonic n-digit Gray codes for any n is based on the idea of recursively building subpaths $P_{n,j}$ of length $2(\genfrac{}{}{0ex}{}{n}{j})$ having edges in $E_n(j)$. We define $P_{1,0}=(\mathtt{0},\mathtt{1})$, $P_{n,j}=\mathrm{\varnothing }$ whenever $j<0$ or $j\ge n$, and

$$P_{n+1,j}=\mathtt{1}{P}_{n,j-1}^{{\pi }_n},\mathtt{0}{P}_{n,j}$$

otherwise. Here, ${\pi }_n$ is a suitably defined permutation and $P^{\pi }$ refers to the path P with its coordinates permuted by $\pi$. These paths give rise to two monotonic n-digit Gray codes $G_n^{(1)}$ and $G_n^{(2)}$ given by

$$G_n^{(1)}=P_{n,0}{P}_{n,1}^R{P}_{n,2}{P}_{n,3}^R\cdots \text{ and }{G}_n^{(2)}=P_{n,0}^R{P}_{n,1}{P}_{n,2}^R{P}_{n,3}\cdots$$

The choice of ${\pi }_n$ which ensures that these codes are indeed Gray codes turns out to be ${\pi }_n=E^{-1}\left({\pi }_{n-1}^2\right)$. The first few values of $P_{n,j}$ are shown in the table below.

Subpaths in the Savage–Winkler algorithm

$P_{n,j}$	j = 0	j = 1	j = 2	j = 3
n = 1	0, 1
n = 2	00, 01	10, 11
n = 3	000, 001	100, 110, 010, 011	101, 111
n = 4	0000, 0001	1000, 1100, 0100, 0110, 0010, 0011	1010, 1011, 1001, 1101, 0101, 0111	1110, 1111

These monotonic Gray codes can be efficiently implemented in such a way that each subsequent element can be generated in O(n) time. The algorithm is most easily described using coroutines.

Monotonic codes have an interesting connection to the Lovász conjecture, which states that every connected vertex-transitive graph contains a Hamiltonian path. The "middle-level" subgraph $Q_{2n+1}(n)$ is vertex-transitive (that is, its automorphism group is transitive, so that each vertex has the same "local environment" and cannot be differentiated from the others, since we can relabel the coordinates as well as the binary digits to obtain an automorphism) and the problem of finding a Hamiltonian path in this subgraph is called the "middle-levels problem", which can provide insights into the more general conjecture. The question has been answered affirmatively for $n\le 15$, and the preceding construction for monotonic codes ensures a Hamiltonian path of length at least 0.839Template:Px2N, where N is the number of vertices in the middle-level subgraph.

Beckett–Gray code

Another type of Gray code, the Beckett–Gray code, is named for Irish playwright Samuel Beckett, who was interested in symmetry. His play "Quad" features four actors and is divided into sixteen time periods. Each period ends with one of the four actors entering or leaving the stage. The play begins and ends with an empty stage, and Beckett wanted each subset of actors to appear on stage exactly once. Clearly the set of actors currently on stage can be represented by a 4-bit binary Gray code. Beckett, however, placed an additional restriction on the script: he wished the actors to enter and exit so that the actor who had been on stage the longest would always be the one to exit. The actors could then be represented by a first in, first out queue, so that (of the actors onstage) the actor being dequeued is always the one who was enqueued first. Beckett was unable to find a Beckett–Gray code for his play, and indeed, an exhaustive listing of all possible sequences reveals that no such code exists for n = 4. It is known today that such codes do exist for n = 2, 5, 6, 7, and 8, and do not exist for n = 3 or 4. An example of an 8-bit Beckett–Gray code can be found in Donald Knuth's Art of Computer Programming. According to Sawada and Wong, the search space for n = 6 can be explored in 15 hours, and more than solutions for the case n = 7 have been found.

Snake-in-the-box codes

Template:Snakes and coils in the box.svg Snake-in-the-box codes, or snakes, are the sequences of nodes of induced paths in an n-dimensional hypercube graph, and coil-in-the-box codes, or coils, are the sequences of nodes of induced cycles in a hypercube. Viewed as Gray codes, these sequences have the property of being able to detect any single-bit coding error. Codes of this type were first described by William H. Kautz in the late 1950s; since then, there has been much research on finding the code with the largest possible number of codewords for a given hypercube dimension.

Single-track Gray code

Yet another kind of Gray code is the single-track Gray code (STGC) developed by Norman B. Spedding and refined by Hiltgen, Paterson and Brandestini in Single-track Gray Codes (1996). The STGC is a cyclical list of P unique binary encodings of length n such that two consecutive words differ in exactly one position, and when the list is examined as a P × n matrix, each column is a cyclic shift of the first column.

File:Animated Graycode.gif

Animated and color-coded version of the STGC rotor.

The name comes from their use with rotary encoders, where a number of tracks are being sensed by contacts, resulting for each in an output of 0 or 1. To reduce noise due to different contacts not switching at exactly the same moment in time, one preferably sets up the tracks so that the data output by the contacts are in Gray code. To get high angular accuracy, one needs lots of contacts; in order to achieve at least 1° accuracy, one needs at least 360 distinct positions per revolution, which requires a minimum of 9 bits of data, and thus the same number of contacts.

If all contacts are placed at the same angular position, then 9 tracks are needed to get a standard BRGC with at least 1° accuracy. However, if the manufacturer moves a contact to a different angular position (but at the same distance from the center shaft), then the corresponding "ring pattern" needs to be rotated the same angle to give the same output. If the most significant bit (the inner ring in Figure 1) is rotated enough, it exactly matches the next ring out. Since both rings are then identical, the inner ring can be cut out, and the sensor for that ring moved to the remaining, identical ring (but offset at that angle from the other sensor on that ring). Those two sensors on a single ring make a quadrature encoder. That reduces the number of tracks for a "1° resolution" angular encoder to 8 tracks. Reducing the number of tracks still further cannot be done with BRGC.

For many years, Torsten Sillke and other mathematicians believed that it was impossible to encode position on a single track such that consecutive positions differed at only a single sensor, except for the 2-sensor, 1-track quadrature encoder. So for applications where 8 tracks were too bulky, people used single-track incremental encoders (quadrature encoders) or 2-track "quadrature encoder + reference notch" encoders.

Norman B. Spedding, however, registered a patent in 1994 with several examples showing that it was possible. Although it is not possible to distinguish 2ⁿ positions with n sensors on a single track, it is possible to distinguish close to that many. Etzion and Paterson conjecture that when n is itself a power of 2, n sensors can distinguish at most 2ⁿ − 2n positions and that for prime n the limit is 2ⁿ − 2 positions. The authors went on to generate a 504-position single track code of length 9 which they believe is optimal. Since this number is larger than 2⁸ = 256, more than 8 sensors are required by any code, although a BRGC could distinguish 512 positions with 9 sensors.

An STGC for P = 30 and n = 5 is reproduced here:

Single-track Gray code for 30 positions

Angle	Code		Angle	Code		Angle	Code		Angle	Code		Angle	Code
0°	10000		72°	01000		144°	00100		216°	00010		288°	00001
12°	10100		84°	01010		156°	00101		228°	10010		300°	01001
24°	11100		96°	01110		168°	00111		240°	10011		312°	11001
36°	11110		108°	01111		180°	10111		252°	11011		324°	11101
48°	11010		120°	01101		192°	10110		264°	01011		336°	10101
60°	11000		132°	01100		204°	00110		276°	00011		348°	10001

Each column is a cyclic shift of the first column, and from any row to the next row only one bit changes. The single-track nature (like a code chain) is useful in the fabrication of these wheels (compared to BRGC), as only one track is needed, thus reducing their cost and size. The Gray code nature is useful (compared to chain codes, also called De Bruijn sequences), as only one sensor will change at any one time, so the uncertainty during a transition between two discrete states will only be plus or minus one unit of angular measurement the device is capable of resolving.

File:9-bit, Single-Track Gray Code.gif

9-bit, single-track Gray code, displaying one degree angular resolution.

Since this 30 degree example was added, there has been a lot of interest in examples with higher angular resolution. In 2008, Gary Williams,Template:Ugc based on previous work, discovered a 9-bit single track Gray code that gives a 1 degree resolution. This Gray code was used to design an actual device which was published on the site Thingiverse. This device was designed by etzenseep (Florian Bauer) in September 2022.

An STGC for P = 360 and n = 9 is reproduced here:

Single-track Gray code for 360 positions

Angle	Code		Angle	Code		Angle	Code		Angle	Code		Angle	Code		Angle	Code		Angle	Code		Angle	Code		Angle	Code
0°	100000001		40°	000000011		80°	000000110		120°	000001100		160°	000011000		200°	000110000		240°	001100000		280°	011000000		320°	110000000
1°	110000001		41°	100000011		81°	000000111		121°	000001110		161°	000011100		201°	000111000		241°	001110000		281°	011100000		321°	111000000
2°	111000001		42°	110000011		82°	100000111		122°	000001111		162°	000011110		202°	000111100		242°	001111000		282°	011110000		322°	111100000
3°	111000011		43°	110000111		83°	100001111		123°	000011111		163°	000111110		203°	001111100		243°	011111000		283°	111110000		323°	111100001
4°	111000111		44°	110001111		84°	100011111		124°	000111111		164°	001111110		204°	011111100		244°	111111000		284°	111110001		324°	111100011
5°	111001111		45°	110011111		85°	100111111		125°	001111111		165°	011111110		205°	111111100		245°	111111001		285°	111110011		325°	111100111
6°	111011111		46°	110111111		86°	101111111		126°	011111111		166°	111111110		206°	111111101		246°	111111011		286°	111110111		326°	111101111
7°	111011011		47°	110110111		87°	101101111		127°	011011111		167°	110111110		207°	101111101		247°	011111011		287°	111110110		327°	111101101
8°	101011011		48°	010110111		88°	101101110		128°	011011101		168°	110111010		208°	101110101		248°	011101011		288°	111010110		328°	110101101
9°	101011111		49°	010111111		89°	101111110		129°	011111101		169°	111111010		209°	111110101		249°	111101011		289°	111010111		329°	110101111
10°	101011101		50°	010111011		90°	101110110		130°	011101101		170°	111011010		210°	110110101		250°	101101011		290°	011010111		330°	110101110
11°	101010101		51°	010101011		91°	101010110		131°	010101101		171°	101011010		211°	010110101		251°	101101010		291°	011010101		331°	110101010
12°	101010111		52°	010101111		92°	101011110		132°	010111101		172°	101111010		212°	011110101		252°	111101010		292°	111010101		332°	110101011
13°	101110111		53°	011101111		93°	111011110		133°	110111101		173°	101111011		213°	011110111		253°	111101110		293°	111011101		333°	110111011
14°	001110111		54°	011101110		94°	111011100		134°	110111001		174°	101110011		214°	011100111		254°	111001110		294°	110011101		334°	100111011
15°	001010111		55°	010101110		95°	101011100		135°	010111001		175°	101110010		215°	011100101		255°	111001010		295°	110010101		335°	100101011
16°	001011111		56°	010111110		96°	101111100		136°	011111001		176°	111110010		216°	111100101		256°	111001011		296°	110010111		336°	100101111
17°	001011011		57°	010110110		97°	101101100		137°	011011001		177°	110110010		217°	101100101		257°	011001011		297°	110010110		337°	100101101
18°	001011001		58°	010110010		98°	101100100		138°	011001001		178°	110010010		218°	100100101		258°	001001011		298°	010010110		338°	100101100
19°	001111001		59°	011110010		99°	111100100		139°	111001001		179°	110010011		219°	100100111		259°	001001111		299°	010011110		339°	100111100
20°	001111101		60°	011111010		100°	111110100		140°	111101001		180°	111010011		220°	110100111		260°	101001111		300°	010011111		340°	100111110
21°	000111101		61°	001111010		101°	011110100		141°	111101000		181°	111010001		221°	110100011		261°	101000111		301°	010001111		341°	100011110
22°	000110101		62°	001101010		102°	011010100		142°	110101000		182°	101010001		222°	010100011		262°	101000110		302°	010001101		342°	100011010
23°	000100101		63°	001001010		103°	010010100		143°	100101000		183°	001010001		223°	010100010		263°	101000100		303°	010001001		343°	100010010
24°	000101101		64°	001011010		104°	010110100		144°	101101000		184°	011010001		224°	110100010		264°	101000101		304°	010001011		344°	100010110
25°	000101001		65°	001010010		105°	010100100		145°	101001000		185°	010010001		225°	100100010		265°	001000101		305°	010001010		345°	100010100
26°	000111001		66°	001110010		106°	011100100		146°	111001000		186°	110010001		226°	100100011		266°	001000111		306°	010001110		346°	100011100
27°	000110001		67°	001100010		107°	011000100		147°	110001000		187°	100010001		227°	000100011		267°	001000110		307°	010001100		347°	100011000
28°	000010001		68°	000100010		108°	001000100		148°	010001000		188°	100010000		228°	000100001		268°	001000010		308°	010000100		348°	100001000
29°	000011001		69°	000110010		109°	001100100		149°	011001000		189°	110010000		229°	100100001		269°	001000011		309°	010000110		349°	100001100
30°	000001001		70°	000010010		110°	000100100		150°	001001000		190°	010010000		230°	100100000		270°	001000001		310°	010000010		350°	100000100
31°	100001001		71°	000010011		111°	000100110		151°	001001100		191°	010011000		231°	100110000		271°	001100001		311°	011000010		351°	110000100
32°	100001101		72°	000011011		112°	000110110		152°	001101100		192°	011011000		232°	110110000		272°	101100001		312°	011000011		352°	110000110
33°	100000101		73°	000001011		113°	000010110		153°	000101100		193°	001011000		233°	010110000		273°	101100000		313°	011000001		353°	110000010
34°	110000101		74°	100001011		114°	000010111		154°	000101110		194°	001011100		234°	010111000		274°	101110000		314°	011100001		354°	111000010
35°	010000101		75°	100001010		115°	000010101		155°	000101010		195°	001010100		235°	010101000		275°	101010000		315°	010100001		355°	101000010
36°	010000111		76°	100001110		116°	000011101		156°	000111010		196°	001110100		236°	011101000		276°	111010000		316°	110100001		356°	101000011
37°	010000011		77°	100000110		117°	000001101		157°	000011010		197°	000110100		237°	001101000		277°	011010000		317°	110100000		357°	101000001
38°	010000001		78°	100000010		118°	000000101		158°	000001010		198°	000010100		238°	000101000		278°	001010000		318°	010100000		358°	101000000
39°	000000001		79°	000000010		119°	000000100		159°	000001000		199°	000010000		239°	000100000		279°	001000000		319°	010000000		359°	100000000

Starting and ending angles for the 20 tracks for a single-track Gray code with 9 sensors separated by 40°

Starting angle	Ending angle	Length
3	4	2
23	28	6
31	37	7
44	48	5
56	60	5
64	71	8
74	76	3
88	91	4
94	96	3
99	104	6
110	115	6
131	134	4
138	154	17
173	181	9
186	187	2
220	238	19
242	246	5
273	279	7
286	289	4
307	360	54

Two-dimensional Gray code

File:16QAM Gray Coded.svg

A Gray-coded constellation diagram for rectangular 16-QAM

Two-dimensional Gray codes are used in communication to minimize the number of bit errors in quadrature amplitude modulation (QAM) adjacent points in the constellation. In a typical encoding the horizontal and vertical adjacent constellation points differ by a single bit, and diagonal adjacent points differ by 2 bits.

Two-dimensional Gray codes also have uses in location identifications schemes, where the code would be applied to area maps such as a Mercator projection of the earth's surface and an appropriate cyclic two-dimensional distance function such as the Mannheim metric be used to calculate the distance between two encoded locations, thereby combining the characteristics of the Hamming distance with the cyclic continuation of a Mercator projection.

Excess Gray code

If a subsection of a specific codevalue is extracted from that value, for example the last 3 bits of a 4-bit Gray code, the resulting code will be an "excess Gray code". This code shows the property of counting backwards in those extracted bits if the original value is further increased. Reason for this is that Gray-encoded values do not show the behaviour of overflow, known from classic binary encoding, when increasing past the "highest" value.

Example: The highest 3-bit Gray code, 7, is encoded as (0)100. Adding 1 results in number 8, encoded in Gray as 1100. The last 3 bits do not overflow and count backwards if you further increase the original 4 bit code.

When working with sensors that output multiple, Gray-encoded values in a serial fashion, one should therefore pay attention whether the sensor produces those multiple values encoded in 1 single Gray code or as separate ones, as otherwise the values might appear to be counting backwards when an "overflow" is expected.

Gray isometry

The bijective mapping { 0 ↔ 00, 1 ↔ 01, 2 ↔ 11, 3 ↔ 10 } establishes an isometry between the metric space over the finite field ${\mathbb{\mathbb{Z}}}_2^2$ with the metric given by the Hamming distance and the metric space over the finite ring ${\mathbb{\mathbb{Z}}}_4$ (the usual modular arithmetic) with the metric given by the Lee distance. The mapping is suitably extended to an isometry of the Hamming spaces ${\mathbb{\mathbb{Z}}}_2^{2m}$ and ${\mathbb{\mathbb{Z}}}_4^m$. Its importance lies in establishing a correspondence between various "good" but not necessarily linear codes as Gray-map images in ${\mathbb{\mathbb{Z}}}_2^2$ of ring-linear codes from ${\mathbb{\mathbb{Z}}}_4$.

Related codes

There are a number of binary codes similar to Gray codes, including:

• Datex codes or Giannini codes (1954), as described by Carl P. Spaulding, use a variant of O'Brien code II.
• Codes used by Varec (c. 1954), use a variant of O'Brien code I as well as base-12 and base-16 Gray code variants.
• Lucal code (1959) aka modified reflected binary code (MRB)
• Gillham code (1961/1962), uses a variant of Datex code and O'Brien code II.
• Leslie and Russell code (1964)
• Royal Radar Establishment code
• Hoklas code (1988)

The following binary-coded decimal (BCD) codes are Gray code variants as well:

• Petherick code (1953), also known as Royal Aircraft Establishment (RAE) code.
• O'Brien codes I and II (1955) (An O'Brien type-I code was already described by Frederic A. Foss of IBM and used by Varec in 1954. Later, it was also known as Watts code or Watts reflected decimal (WRD) code and is sometimes ambiguously referred to as reflected binary modified Gray code. An O'Brien type-II code was already used by Datex in 1954.)
• Excess-3 Gray code (1956) (aka Gray excess-3 code, Gray 3-excess code, reflex excess-3 code, excess Gray code, Gray excess code, 10-excess-3 Gray code or Gray–Stibitz code), described by Frank P. Turvey Jr. of ITT.
• Tompkins codes I and II (1956)
• Glixon code (1957), sometimes ambiguously also called modified Gray code

4-bit unit-distance BCD codes

Name	Bit	0	1	2	3	4	5	6	7	8	9	Weights	Tracks	Compl.	Cyclic	5s	Comment
Gray BCD	4	0	0	0	0	0	0	0	0	1	1	0–3	4 (3)	No	(2, 4, 8, 16)	No
	3	0	0	0	0	1	1	1	1	1	1
	2	0	0	1	1	1	1	0	0	0	0
	1	0	1	1	0	0	1	1	0	0	1
Paul	4	1	0	0	0	0	0	0	0	1	1	1–3	4 (3)	No	2, 10	No
	3	0	0	0	0	1	1	1	1	1	1
	2	0	0	1	1	1	1	0	0	0	0
	1	1	1	1	0	0	1	1	0	0	1
Glixon	4	0	0	0	0	0	0	0	0	1	1	0–3	4	No	2, 4, 8, 10	(shifted +1)
	3	0	0	0	0	1	1	1	1	1	0
	2	0	0	1	1	1	1	0	0	0	0
	1	0	1	1	0	0	1	1	0	0	0
Tompkins I	4	0	0	0	0	0	1	1	1	1	1	0–4	2	No	2, 4, 10	Yes
	3	0	0	0	0	1	1	1	1	1	0
	2	0	0	1	1	1	1	1	0	0	0
	1	0	1	1	0	0	0	1	1	0	0
O'Brien I (Watts)	4	0	0	0	0	0	1	1	1	1	1	0–3	4	9	2, 4, 10	Yes
	3	0	0	0	0	1	1	0	0	0	0
	2	0	0	1	1	1	1	1	1	0	0
	1	0	1	1	0	0	0	0	1	1	0
Petherick (RAE)	4	0	0	0	0	0	1	1	1	1	1	1–3	3	9	2, 10	Yes
	3	1	0	0	0	1	1	0	0	0	1
	2	0	0	1	1	1	1	1	1	0	0
	1	1	1	1	0	0	0	0	1	1	1
O'Brien II	4	0	0	0	0	0	1	1	1	1	1	1–3	3	9	2, 10	Yes
	3	0	0	0	1	1	1	1	0	0	0
	2	0	1	1	1	0	0	1	1	1	0
	1	1	1	0	0	0	0	0	0	1	1
Susskind	4	0	0	0	0	0	1	1	1	1	1	1–4	3	9	2, 10	Yes
	3	0	0	1	1	1	1	1	1	0	0
	2	0	1	1	1	0	0	1	1	1	0
	1	1	1	1	0	0	0	0	1	1	1
Klar	4	0	0	0	0	0	1	1	1	1	1	0–4	4 (3)	9	2, 10	Yes
	3	0	0	0	1	1	1	1	0	0	0
	2	0	0	1	1	1	1	1	1	0	0
	1	0	1	1	1	0	0	1	1	1	0
Tompkins II	4	0	0	0	0	0	1	1	1	1	1	1–3	2	9	2, 10	Yes
	3	0	0	1	1	1	1	1	0	0	0
	2	1	1	1	0	0	0	0	0	1	1
	1	0	1	1	1	0	0	1	1	1	0
Excess-3 Gray	4	0	0	0	0	0	1	1	1	1	1	1–4	4	9	2, 10	Yes
	3	0	1	1	1	1	1	1	1	1	0
	2	1	1	1	0	0	0	0	1	1	1
	1	0	0	1	1	0	0	1	1	0	0

See also

• Linear-feedback shift register
• De Bruijn sequence
• Steinhaus–Johnson–Trotter algorithm – an algorithm that generates Gray codes for the factorial number system
• Minimum distance code
• Prouhet–Thue–Morse sequence – related to inverse Gray code
• Ryser formula
• Hilbert curve

Further reading

• Part 2 Part 3
• (7 pages)
• (5 pages)
• (2 pages)

External links

• "Gray Code" demonstration by Michael Schreiber, Wolfram Demonstrations Project (with Mathematica implementation). 2007.
• NIST Dictionary of Algorithms and Data Structures: Gray code.
• Hitch Hiker's Guide to Evolutionary Computation, Q21: What are Gray codes, and why are they used?, including C code to convert between binary and BRGC.
• Dragos A. Harabor uses Gray codes in a 3D digitizer.
• Single-track gray codes, binary chain codes (Lancaster 1994), and linear-feedback shift registers are all useful in finding one's absolute position on a single-track rotary encoder (or other position sensor).
• AMS Column: Gray codes
• Optical Encoder Wheel Generator
• ProtoTalk.net – Understanding Quadrature Encoding – Covers quadrature encoding in more detail with a focus on robotic applications