Gray code
The use of Gray code in these devices helps simplify logic operations and reduce errors in practice.
[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.
[31] Martin Gardner wrote a popular account of the Gray code in his August 1972 "Mathematical Games" column in Scientific American.
[40][41][38] About the same time, the German-Austrian Otto Schäffler [de][42] demonstrated another printing telegraph in Vienna using a 5-bit reflected binary code for the same purpose, in 1874.
For example, some rotary encoders provide a disk which has an electrically conductive Gray code pattern on concentric rings (tracks).
Other encoders employ non-contact mechanisms based on optical or magnetic sensors to produce the Gray code output signals.
A binary output code could cause significant position measurement errors because it is impossible to make all bits change at exactly the same time.
[15] 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.
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.
[11][12][13] 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.
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.
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.
[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 master-slave flip flops in a binary ripple counter.
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 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
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.
In Guan's algorithm, the count alternately rises and falls, so that the numeric difference between two Gray code digits is always one.
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.
[52] In balanced Gray codes, the number of changes in different coordinate positions are as close as possible.
[66] Long run (or maximum gap) Gray codes maximize the distance between consecutive changes of digits in the same position.
[68] Monotonic codes are useful in the theory of interconnection networks, especially for minimizing dilation for linear arrays of processors.
An elegant construction of monotonic n-digit Gray codes for any n is based on the idea of recursively building subpaths
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.
, and the preceding construction for monotonic codes ensures a Hamiltonian path of length at least 0.839N, where N is the number of vertices in the middle-level subgraph.
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.
[81] 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.
based on previous work,[80] discovered a 9-bit single track Gray code that gives a 1 degree resolution.
An STGC for P = 360 and n = 9 is reproduced here: 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.
This code shows the property of counting backwards in those extracted bits if the original value is further increased.
