Lucal code^{[1]}^{[2]}  

5  4  3  2  1  
Gray code  
4  3  2  1  
0  0  0  0  0  0 
1  0  0  0  1  1 
2  0  0  1  1  0 
3  0  0  1  0  1 
4  0  1  1  0  0 
5  0  1  1  1  1 
6  0  1  0  1  0 
7  0  1  0  0  1 
8  1  1  0  0  0 
9  1  1  0  1  1 
10  1  1  1  1  0 
11  1  1  1  0  1 
12  1  0  1  0  0 
13  1  0  1  1  1 
14  1  0  0  1  0 
15  1  0  0  0  1 
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.^{[3]}
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 unitdistance,^{[4]}^{[5]}^{[6]}^{[7]}^{[8]} singledistance, singlestep, monostrophic^{[9]}^{[10]}^{[7]}^{[8]} or syncopic codes,^{[9]} in reference to the Hamming distance of 1 between adjacent codes.
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 nonnegative integers, the binaryreflected Gray code, or BRGC. Bell Labs researcher George R. Stibitz described such a code in a 1941 patent application, granted in 1943.^{[11]}^{[12]}^{[13]} Frank Gray introduced the term reflected binary code in his 1947 patent application, remarking that the code had "as yet no recognized name".^{[14]} He derived the name from the fact that it "may be built up from the conventional binary code by a sort of reflection process".
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 ith least significant bit a pattern of 2^{i} on 2^{i} off. The most significant digit is an exception to this: for an nbit Gray code, the most significant digit follows the pattern 2^{n1} on, 2^{n1} off, which is the same (cyclic) sequence of values as for the secondmost significant digit, but shifted forwards 2^{n2} places. The fourbit 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.^{[15]}
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 singlebit 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^{[11]}^{[12]}^{[13]} 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";^{[16]}^{[17]} one of those also lists "minimum error code" and "cyclic permutation code" among the names.^{[17]} A 1954 patent application refers to "the Bell Telephone Gray code".^{[18]} Other names include "cyclic binary code",^{[12]} "cyclic progression code",^{[19]}^{[12]} "cyclic permuting binary"^{[20]} or "cyclic permuted binary" (CPB).^{[21]}^{[22]}
The Gray code is sometimes misattributed to 19th century electrical device inventor Elisha Gray.^{[13]}^{[23]}^{[24]}^{[25]}
Reflected binary codes were applied to mathematical puzzles before they became known to engineers.
The binaryreflected 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.^{[26]}^{[13]}
It can serve as a solution guide for the Towers of Hanoi problem, based on a game by the French Édouard Lucas in 1883.^{[27]}^{[28]}^{[29]}^{[30]} Similarly, the socalled Towers of Bucharest and Towers of Klagenfurt game configurations yield ternary and pentary Gray codes.^{[31]}
Martin Gardner wrote a popular account of the Gray code in his August 1972 Mathematical Games column in Scientific American.^{[32]}
The code also forms a Hamiltonian cycle on a hypercube, where each bit is seen as one dimension.
When the French engineer Émile Baudot changed from using a 6unit (6bit) code to 5unit code for his printing telegraph system, in 1875^{[33]} or 1876,^{[34]}^{[35]} 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,^{[36]}^{[37]}^{[38]} and other symbols appropriately placed, the 5bit character code has been recognized as a reflected binary code.^{[13]} This code became known as Baudot code^{[39]} and, with minor changes, was eventually adopted as International Telegraph Alphabet No. 1 (ITA1, CCITT1) in 1932.^{[40]}^{[41]}^{[38]}
About the same time, the GermanAustrian Otto Schäffler
^{[42]} demonstrated another printing telegraph in Vienna using a 5bit reflected binary code for the same purpose, in 1874.^{[43]}^{[13]}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 tubebased apparatus. Filed in 1947, the method and apparatus were granted a patent in 1953,^{[14]} 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.^{[44]}
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.
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 noncontact 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.
Due to the Hamming distance properties of Gray codes, they are sometimes used in genetic algorithms.^{[15]} They are very useful in this field, since mutations in the code allow for mostly incremental changes, but occasionally a single bitchange can cause a big leap and lead to new properties.
Gray codes are also used in labelling the axes of Karnaugh maps since 1953^{[45]}^{[46]}^{[47]} as well as in Händler circle graphs since 1958,^{[48]}^{[49]}^{[50]}^{[51]} both graphical methods for logic circuit minimization.
In modern digital communications, 1D and 2DGray 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 singlebit 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.
Digital logic designers use Gray codes extensively for passing multibit 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.
If a system has to cycle sequentially through all possible combinations of onoff states of some set of controls, and the changes of the controls require nontrivial 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,^{[52]} that flips every bit equally often. Since bitflips are evenly distributed, this is optimal in the following way: balanced Gray codes minimize the maximal count of bitflips for each digit.
George R. Stibitz utilized a reflected binary code in a binary pulse counting device in 1941 already.^{[11]}^{[12]}^{[13]}
A typical use of Gray code counters is building a FIFO (firstin, firstout) data buffer that has read and write ports that exist in different clock domains. The input and output counters inside such a dualport FIFO are often stored using Gray code to prevent invalid transient states from being captured when the count crosses clock domains.^{[53]} 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 nondeterministically 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 multibit 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 multibit value. Typically Gray codes of poweroftwo 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 digitaltoanalog 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,^{[nb 1]} it is possible that differences in the arrival times of the binary data bits into the binarytoGray 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.^{[54]}
To produce the next count value in a Graycode 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,^{[55]} add one to it with a standard binary adder, and then convert the result back to Gray code.^{[56]} 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 masterslave flip flops in a binary ripple counter.^{[57]}
As the execution of program 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 lowpower designs.^{[58]}^{[59]}
The binaryreflected 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.^{[13]} For example, generating the n = 3 list from the n = 2 list:
2bit 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 onebit Gray code is G_{1} = (0,1). This can be thought of as built recursively as above from a zerobit Gray code G_{0} = ( 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:
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 bitshift and exclusiveor operation if they are available: the nth Gray code is obtained by computing . 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 of codewords are inverted, the order of neighbouring blocks of 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:
If bit 2 is inverted, blocks of 4 codewords reverse order:
Thus, performing an exclusive or on a bit at position with the bit at position leaves the order of codewords intact if , and reverses the order of blocks of codewords if . Now, this is exactly the same operation as the reflectandprefix 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 is the th Graycoded bit ( being the most significant bit), and is the th binarycoded bit ( being the mostsignificant bit), the reverse translation can be given recursively: , and . 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 binaryreflected Gray code iteratively, at step 0 start with the , and at step find the bit position of the least significant 1 in the binary representation of and flip the bit at that position in the previous code to get the next code . The bit positions start 0, 1, 0, 2, 0, 1, 0, 3, ....^{[nb 2]} See find first set for efficient algorithms to compute these values.
The following functions in C convert between binary numbers and their associated Gray codes. While it may seem that Graytobinary conversion requires each bit to be handled one at a time, faster algorithms exist.^{[60]}^{[55]}^{[nb 1]}
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 exclusiveored 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 Fourbitatonce 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.
In practice, "Gray code" almost always refers to a binaryreflected 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).
It is possible to construct binary Gray codes with n bits with a length of less than 2^{n}, 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.^{[61]} OEIS sequence A290772 ^{[62]} gives the number of possible Gray sequences of length 2n that include zero and use the minimum number of bits.

There are many specialized types of Gray codes other than the binaryreflected Gray code. One such type of Gray code is the nary Gray code, also known as a nonBoolean Gray code. As the name implies, this type of Gray code uses nonBoolean values in its encodings.
For example, a 3ary (ternary) Gray code would use the values 0,1,2.^{[31]} The (n, k)Gray code is the nary Gray code with k digits.^{[63]} 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 baseN 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,^{[63]} 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.
Although the binary reflected Gray code is useful in many scenarios, it is not optimal in certain cases because of a lack of "uniformity".^{[52]} 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 Rary complete Gray cycle having transition sequence ; the transition counts (spectrum) of G are the collection of integers defined by
A Gray code is uniform or uniformly balanced if its transition counts are all equal, in which case we have for all k. Clearly, when , such codes exist only if n is a power of 2.^{[64]} If n is not a power of 2, it is possible to construct wellbalanced binary codes where the difference between two transition counts is at most 2; so that (combining both cases) every transition count is either or .^{[52]} 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.^{[65]}
For example, a balanced 4bit Gray code has 16 transitions, which can be evenly distributed among all four positions (four transitions per position), making it uniformly balanced:^{[52]}
whereas a balanced 5bit 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:^{[52]}
We will now show a construction^{[66]} and implementation^{[67]} for wellbalanced binary Gray codes which allows us to generate an ndigit balanced Gray code for every n. The main principle is to inductively construct an (n + 2)digit Gray code given an ndigit Gray code G in such a way that the balanced property is preserved. To do this, we consider partitions of into an even number L of nonempty blocks of the form
where , , and ). This partition induces an digit Gray code given by
If we define the transition multiplicities
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 is
The delicate part of this construction is to find an adequate partitioning of a balanced ndigit 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 transition and splitting another block at another digit transition produces a different Gray code with exactly the same transition spectrum , so one may for example^{[65]} designate the first transitions at digit as those that fall between two blocks. Uniform codes can be found when and , and this construction can be extended to the Rary case as well.^{[66]}
Long run (or maximum gap) Gray codes maximize the distance between consecutive changes of digits in the same position. That is, the minimum runlength of any bit remains unchanged for as long as possible.^{[68]}
Monotonic codes are useful in the theory of interconnection networks, especially for minimizing dilation for linear arrays of processors.^{[69]} 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 into levels of vertices that have equal weight, i.e.
for . These levels satisfy . Let be the subgraph of induced by , and let be the edges in . A monotonic Gray code is then a Hamiltonian path in such that whenever comes before in the path, then .
An elegant construction of monotonic ndigit Gray codes for any n is based on the idea of recursively building subpaths of length having edges in .^{[69]} We define , whenever or , and
otherwise. Here, is a suitably defined permutation and refers to the path P with its coordinates permuted by . These paths give rise to two monotonic ndigit Gray codes and given by
The choice of which ensures that these codes are indeed Gray codes turns out to be . The first few values of are shown in the table below.
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 vertextransitive graph contains a Hamiltonian path. The "middlelevel" subgraph is vertextransitive (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 "middlelevels problem", which can provide insights into the more general conjecture. The question has been answered affirmatively for , and the preceding construction for monotonic codes ensures a Hamiltonian path of length at least 0.839 N, where N is the number of vertices in the middlelevel subgraph.^{[70]}
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.^{[71]} Clearly the set of actors currently on stage can be represented by a 4bit 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.^{[71]} 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 8bit Beckett–Gray code can be found in Donald Knuth's Art of Computer Programming.^{[13]} According to Sawada and Wong, the search space for n = 6 can be explored in 15 hours, and more than 9500 solutions for the case n = 7 have been found.^{[72]}
Snakeinthebox codes, or snakes, are the sequences of nodes of induced paths in an ndimensional hypercube graph, and coilinthebox codes,^{[73]} 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 singlebit coding error. Codes of this type were first described by William H. Kautz in the late 1950s;^{[5]} since then, there has been much research on finding the code with the largest possible number of codewords for a given hypercube dimension.
Yet another kind of Gray code is the singletrack Gray code (STGC) developed by Norman B. Spedding^{[74]}^{[75]} and refined by Hiltgen, Paterson and Brandestini in Singletrack Gray Codes (1996).^{[76]}^{[77]} 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.^{[78]}
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^{[79]} 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 2sensor, 1track quadrature encoder. So for applications where 8 tracks were too bulky, people used singletrack incremental encoders (quadrature encoders) or 2track "quadrature encoder + reference notch" encoders.
Norman B. Spedding, however, registered a patent in 1994 with several examples showing that it was possible.^{[74]} Although it is not possible to distinguish 2^{n} 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 2^{n} − 2n positions and that for prime n the limit is 2^{n} − 2 positions.^{[80]} The authors went on to generate a 504position single track code of length 9 which they believe is optimal. Since this number is larger than 2^{8} = 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:
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.^{[81]} The singletrack 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.^{[82]}
Since this 30 degree example was added, there has been a lot of interest in examples with higher angular resolution. In 2008, Gary Williams,^{[83]} based on previous work ^{[80]} discovered a 9bit 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^{[84]} was designed by etzenseep (Florian Bauer) in September, 2022.
An STGC for P = 360 and n = 9 is reproduced here:
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 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 
Twodimensional 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.^{[85]}
Twodimensional 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 twodimensional 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.^{[86]}
If a subsection of a specific codevalue is extracted from that value, for example the last 3 bits of a 4bit graycode, 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 grayencoded values do not show the behaviour of overflow, known from classic binary encoding, when increasing past the "highest" value.
Example: The highest 3bit 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, grayencoded values in a serial fashion, one should therefore pay attention whether the sensor produces those multiple values encoded in 1 single graycode or as separate ones, as otherwise the values might appear to be counting backwards when an "overflow" is expected.
The bijective mapping { 0 ↔ 00, 1 ↔ 01, 2 ↔ 11, 3 ↔ 10 } establishes an isometry between the metric space over the finite field with the metric given by the Hamming distance and the metric space over the finite ring (the usual modular arithmetic) with the metric given by the Lee distance. The mapping is suitably extended to an isometry of the Hamming spaces and . Its importance lies in establishing a correspondence between various "good" but not necessarily linear codes as Graymap images in of ringlinear codes from .^{[87]}^{[88]}
There are a number of binary codes similar to Gray codes, including:
The following binarycoded decimal (BCD) codes are Gray code variants as well:
Name  Bit  0  1  2  3  4  5  6  7  8  9  Weights^{[nb 7]}  Tracks  Compl.  Cyclic  5s  Comment 
Gray BCD  4  0  0  0  0  0  0  0  0  1  1  0—3  4 (3^{[nb 8]})  No  (2, 4, 8, 16)  No  ^{[110]}^{[111]} 
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^{[nb 8]})  No  2, 10  No  ^{[125]} 
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)  ^{[122]}^{[110]}^{[111]}^{[123]}^{[124]}^{[nb 5]} 
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  ^{[4]}^{[110]}^{[111]} 
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^{[103]}^{[104]}^{[nb 9]}  2, 4, 10  Yes  ^{[109]}^{[110]}^{[111]}^{[nb 5]} 
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^{[103]}^{[104]}^{[nb 9]}  2, 10  Yes  ^{[19]}^{[107]}^{[nb 4]} 
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^{[91]}^{[103]}^{[104]}^{[nb 9]}  2, 10  Yes  ^{[109]}^{[110]}^{[111]}^{[nb 4]} 
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^{[nb 9]}  2, 10  Yes  ^{[6]} 
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^{[nb 8]})  9^{[nb 9]}  2, 10  Yes  ^{[126]}^{[127]} 
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^{[nb 10]}  2, 10  Yes  ^{[4]}^{[110]}^{[111]} 
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  
Excess3 Gray  4  0  0  0  0  0  1  1  1  1  1  1—4  4  9^{[103]}^{[104]}^{[nb 9]}  2, 10  Yes  ^{[8]}^{[103]} 
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 
[…] A clearer idea of the position of the balls after each pulse will be obtained if the set of balls is represented by a number having a similar number of digits, each of which may have one of two arbitrary values, for example 0 and 1. If the upper position is called 0 and the lower position […] 1, then the setting of the counter […] may be read from left to right as 0,100,000. […] Following is a translation of the number of pulses received into this form of binary notation for the first sixteen pulses as received on the first five balls […] Pulse number […] Binary notation […][1] (4 pages)
[…] The type of code wheel most popular in optical encoders contains a cyclic binary code pattern designed to give a cyclic sequence of "onoff" outputs. The cyclic binary code is also known as the cyclic progression code, the reflected binary code, and the Gray code. This code was originated by G. R. Stibitz, of Bell Telephone Laboratories, and was first proposed for pulsecode modulation systems by Frank Gray, also of BTL. Thus the name Gray code. The Gray or cyclic code is used mainly to eliminate the possibility of errors at code transition which could result in gross ambiguities. […]
[…] Decoding. […] To decode C.P.B. or W.R.D. codes, a simple inversion rule can be applied. The readings of the higher tracks determine the way in which the lower tracks are translated. The inversion rule is applied line by line for the C.P.B. and for the W.R.D. it is applied decade by decade or line by line. Starting therefore with the top or slowest changing track of the C.P.B., if the result is odd (1) the next track value has to be inverted, i.e. 0 for 1 and 1 for 0. If, however, the first track is even (0), the second track is left as read, i.e. 0 for 0 and 1 for 1. Again, if the resultant reading of the second track is odd, the third track reading is inverted and so on. When an odd is changed to an even the line below is not inverted and when an even is changed to an odd the line below is inverted. The result of applying this rule to the pattern […] is the pure binary (P.B.) pattern […] where each track or digit can be given a definite numerical value (in this instance 1, 2, 4, 8, etc.). […] Using the linebyline inversion rule on the W.R.D. code produces [a] pattern [of 1, 2, 4, 2 code] where again the digits can be given numerical values and summed decade by decade. The summing of the digits can be very useful, for example, in a highspeed scanning system; but in a parallel decoding system […], it is usual to treat each binary quartet or decade as an entity. In other words, if the first or more significant decade is odd, the second decade is rectified or complemented by inverting the D track and so on, the result being the repeating pattern of [rectified W.R.D. code]. This is an extremely easy thing to achieve since the only change required is the inversion of the meaning of the D track or complementing digit. […](8+82 pages) (NB. The author does not mention Gray at all and calls the standard Gray code "Cyclic Permuted Binary Code" (C.P.B.), the book index erroneously lists it as "cyclic pure binary code".)
[…] There seems to be some confusion about the attributation of this code, because two inventors named Gray have been associated with it. When I first heard the name I took it as referring to Elisha Gray, and Heath testifies to his usage of it. Many people take it as referring to Frank Gray of Bell Telephone Laboratories, who in 1947 first proposed its use in coding tubes: his patent is listed in the bibliography. […](2+448+2 pages)
{{cite journal}}
: CS1 maint: DOI inactive as of November 2024 (link) [3] (4 pages)
[…] Der um die Mitte des J[ahres] 1874 patenti[e]rte, ebenfalls dem Highton'schen verwandte Typendrucker des französischen TelegraphenVerwaltungsbeamten Baudot wurde bei seiner 1875 patenti[e]rten Weiterentwicklung in einen fünffachen umgewandelt […]
[…] A Baudot prototype (4 years in the making) was built in 1876. The transmitter had 5 keys similar to those of a piano. Messages were sent in a special 5element code devised by Baudot […]
[…] In 1872, [Baudot] started research toward a telegraph system that would allow multiple operators to transmit simultaneously over a single wire and, as the transmissions were received, would print them in ordinary alphabetic characters on a strip of paper. He received a patent for such a system on June 17, 1874. […] Instead of a variable delay followed by a singleunit pulse, Baudot's system used a uniform six time units to transmit each character. […] his early telegraph probably used the sixunit code […] that he attributes to Davy in an 1877 article. […] in 1876 Baudot redesigned his equipment to use a fiveunit code. Punctuation and digits were still sometimes needed, though, so he adopted from Hughes the use of two special letter space and figure space characters that would cause the printer to shift between cases at the same time as it advanced the paper without printing. The fiveunit code he began using at this time […] was structured to suit his keyboard […], which controlled two units of each character with switches operated by the left hand and the other three units with the right hand. […][5][6]
[…] In 1874, Schaefflerprinting telegraph, a quadruple system like the Baudot, but mechanically more sophisticated. The Hughes telegraph had two synchronously rotating fingers, one in the sender and one in the receiver. By a pianolike keyboard the operator selected a letter and thereby made contact with the rotating finger in the corresponding direction. Since the receiving finger was in the same direction at this moment, the receiver could print the correct letter. The Baudot and the Schaeffler printing telegraphs use a fivebit binary code. ... Schaeffler's code is a reflected binary code! What F. Gray patented in 1953 for PCM, Schaeffler had applied in his telegraph in 1874, and for a similar reason: reliability. He had contact fingers sensing on five cams consecutively all combinations; the right one triggers printing. If the fingers are to make a minimal number of movements, the solution is the reflected binary code. For Schaeffler, this idea was a minor one. More exactly, the code is described in a letter by the Austrian Post employee, J[ohann] N[epomuk] Teufelhart, inserted there as a footnote and telling that Schaeffler found the code by combining wooden bars with the different combinations until he had the best solution. Another Post employee, Alexander Wilhelm Lambert of Linz, claims to have shown this code to Schaeffler as early as 1872, but this claim is not clear and cannot be checked. […](6 pages) invented another
[…] Karnaugh's map orders the arguments of the discriminants according to the reflected binary code, also called the Gray code. […](xii+291+3 pages) 1st edition
[…] Übersichtlich ist die Darstellung nach Händler, die sämtliche Punkte, numeriert nach dem GrayCode […], auf dem Umfeld eines Kreises anordnet. Sie erfordert allerdings sehr viel Platz. […][Händler's diagram, where all points, numbered according to the Gray code, are arranged on the circumference of a circle, is easily comprehensible. It needs, however, a lot of space.]
[…] The MOAGILLHAM code is essentially the combination of the Gray code discussed thereinabove and the well known Datex code; the Datex code is disclosed in U.S. Patent 3,165,731. The arrangement is such that the Datex code defines the bits for the units count of the encoder and the Gray code defines the bits for each of the higher order decades, the tens, hundreds, etc. […](11 pages)
[…] The Datex code […] uses the O'Brien code II within each decade, and reflected decimal numbers for the decimal transitions. For further processing, code conversion to the natural decimal notation is necessary. Since the O'Brien II code forms a 9s complement, this does not give rise to particular difficulties: whenever the code word for the tens represents an odd number, the code words for the decimal units are given as the 9s complements by inversion of the fourth binary digit. […]^{[permanent dead link]} (270 pages)
[…] The complete dispatching operation, gauging, and remote control is integrated into one single unitized system when a "Varec" Pulse Code Telemetering System is installed. […]
[…] Other forms of code are also well known. Among these are the Royal Radar Establishment code; The Excess Three decimal code; Gillham code which is recommended by ICAO for automatic height transmission for air traffic control purposes; the Petherick code, and the Leslie and Russell code of the National Engineering Laboratory. Each has its particular merits and they are offered as options by various encoder manufacturers. […](12+367+5 pages)
[…] Die Firma Harrison Reproduction Equipment, Farnborough/England […] hat in jahrelanger Entwicklung in Zusammenarbeit mit der Britischen Luftwaffe und britischen Industriebetrieben den mechanischen Digitizer […] zu einer technischen Reife gebracht, die fast allen Anforderungen […] genügt. […] Um bei der dezimalen Entschlüsselung des verwendeten Binärcodes zu eindeutigen und bei der Übergabe von einer Dezimalstelle zur anderen in der Reihenfolge immer richtigen Ergebnissen zu kommen, wurde ein spezieller Code entwickelt, der jede Möglichkeit einer Fehlaussage durch sein Prinzip ausschließt und der außerdem durch seinen Aufbau eine relativ einfache Entschlüsselung erlaubt. Der Code basiert auf dem PetherickCode. […](4 pages)