|
| 11010}} | 120° | {{mono | | | 10110}} | 264° | {{mono | | | 10101}} |
|
| 11000}} | 132° | {{mono | | | 00110}} | 276° | {{mono | | | 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|thumb|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,WEB, Williams, Gary, ‘single track gray code’ sought for encoding 360 degrees with 9 sensors,www.experts-exchange.com/questions/23594359/’single-track-gray-code’-sought-for-encoding-360-degrees-with-9-sensors.html#a22127642, Experts Exchange, 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 deviceWEB, Bauer, Florian, 9-Bit Absolute Singletrack Gray Code Rotary Encoder,
www.thingiverse.com/thing:4590077, Thingiverse, was designed by etzenseep (Florian Bauer) in September, 2022.
{|class=“wikitable” style="text-align:center; background:#FFFFFF; border-width:0;“|+ 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 | |
| |100000001}} | | |000000011}} | | |000000110}} | | |000001100}} | | |000011000}} | | |000110000}} | | |001100000}} | | |011000000}} | | |110000000}} | |
| |110000001}} | | |100000011}} | | |000000111}} | | |000001110}} | | |000011100}} | | |000111000}} | | |001110000}} | | |011100000}} | | |111000000}} | |
| |111000001}} | | |110000011}} | | |100000111}} | | |000001111}} | | |000011110}} | | |000111100}} | | |001111000}} | | |011110000}} | | |111100000}} | |
| |111000011}} | | |110000111}} | | |100001111}} | | |000011111}} | | |000111110}} | | |001111100}} | | |011111000}} | | |111110000}} | | |111100001}} | |
| |111000111}} | | |110001111}} | | |100011111}} | | |000111111}} | | |001111110}} | | |011111100}} | | |111111000}} | | |111110001}} | | |111100011}} | |
| |111001111}} | | |110011111}} | | |100111111}} | | |001111111}} | | |011111110}} | | |111111100}} | | |111111001}} | | |111110011}} | | |111100111}} | |
| |111011111}} | | |110111111}} | | |101111111}} | | |011111111}} | | |111111110}} | | |111111101}} | | |111111011}} | | |111110111}} | | |111101111}} | |
| |111011011}} | | |110110111}} | | |101101111}} | | |011011111}} | | |110111110}} | | |101111101}} | | |011111011}} | | |111110110}} | | |111101101}} | |
| |101011011}} | | |010110111}} | | |101101110}} | | |011011101}} | | |110111010}} | | |101110101}} | | |011101011}} | | |111010110}} | | |110101101}} | |
| |101011111}} | | |010111111}} | | |101111110}} | | |011111101}} | | |111111010}} | | |111110101}} | | |111101011}} | | |111010111}} | | |110101111}} | |
| |101011101}} | | |010111011}} | | |101110110}} | | |011101101}} | | |111011010}} | | |110110101}} | | |101101011}} | | |011010111}} | | |110101110}} | |
| |101010101}} | | |010101011}} | | |101010110}} | | |010101101}} | | |101011010}} | | |010110101}} | | |101101010}} | | |011010101}} | | |110101010}} | |
| |101010111}} | | |010101111}} | | |101011110}} | | |010111101}} | | |101111010}} | | |011110101}} | | |111101010}} | | |111010101}} | | |110101011}} | |
| |101110111}} | | |011101111}} | | |111011110}} | | |110111101}} | | |101111011}} | | |011110111}} | | |111101110}} | | |111011101}} | | |110111011}} | |
| |001110111}} | | |011101110}} | | |111011100}} | | |110111001}} | | |101110011}} | | |011100111}} | | |111001110}} | | |110011101}} | | |100111011}} | |
| |001010111}} | | |010101110}} | | |101011100}} | | |010111001}} | | |101110010}} | | |011100101}} | | |111001010}} | | |110010101}} | | |100101011}} | |
| |001011111}} | | |010111110}} | | |101111100}} | | |011111001}} | | |111110010}} | | |111100101}} | | |111001011}} | | |110010111}} | | |100101111}} | |
| |001011011}} | | |010110110}} | | |101101100}} | | |011011001}} | | |110110010}} | | |101100101}} | | |011001011}} | | |110010110}} | | |100101101}} | |
| |001011001}} | | |010110010}} | | |101100100}} | | |011001001}} | | |110010010}} | | |100100101}} | | |001001011}} | | |010010110}} | | |100101100}} | |
| |001111001}} | | |011110010}} | | |111100100}} | | |111001001}} | | |110010011}} | | |100100111}} | | |001001111}} | | |010011110}} | | |100111100}} | |
| |001111101}} | | |011111010}} | | |111110100}} | | |111101001}} | | |111010011}} | | |110100111}} | | |101001111}} | | |010011111}} | | |100111110}} | |
| |000111101}} | | |001111010}} | | |011110100}} | | |111101000}} | | |111010001}} | | |110100011}} | | |101000111}} | | |010001111}} | | |100011110}} | |
| |000110101}} | | |001101010}} | | |011010100}} | | |110101000}} | | |101010001}} | | |010100011}} | | |101000110}} | | |010001101}} | | |100011010}} | |
| |000100101}} | | |001001010}} | | |010010100}} | | |100101000}} | | |001010001}} | | |010100010}} | | |101000100}} | | |010001001}} | | |100010010}} | |
| |000101101}} | | |001011010}} | | |010110100}} | | |101101000}} | | |011010001}} | | |110100010}} | | |101000101}} | | |010001011}} | | |100010110}} | |
| |000101001}} | | |001010010}} | | |010100100}} | | |101001000}} | | |010010001}} | | |100100010}} | | |001000101}} | | |010001010}} | | |100010100}} | |
| |000111001}} | | |001110010}} | | |011100100}} | | |111001000}} | | |110010001}} | | |100100011}} | | |001000111}} | | |010001110}} | | |100011100}} | |
| |000110001}} | | |001100010}} | | |011000100}} | | |110001000}} | | |100010001}} | | |000100011}} | | |001000110}} | | |010001100}} | | |100011000}} | |
| |000010001}} | | |000100010}} | | |001000100}} | | |010001000}} | | |100010000}} | | |000100001}} | | |001000010}} | | |010000100}} | | |100001000}} | |
| |000011001}} | | |000110010}} | | |001100100}} | | |011001000}} | | |110010000}} | | |100100001}} | | |001000011}} | | |010000110}} | | |100001100}} | |
| |000001001}} | | |000010010}} | | |000100100}} | | |001001000}} | | |010010000}} | | |100100000}} | | |001000001}} | | |010000010}} | | |100000100}} | |
| |100001001}} | | |000010011}} | | |000100110}} | | |001001100}} | | |010011000}} | | |100110000}} | | |001100001}} | | |011000010}} | | |110000100}} | |
| |100001101}} | | |000011011}} | | |000110110}} | | |001101100}} | | |011011000}} | | |110110000}} | | |101100001}} | | |011000011}} | | |110000110}} | |
| |100000101}} | | |000001011}} | | |000010110}} | | |000101100}} | | |001011000}} | | |010110000}} | | |101100000}} | | |011000001}} | | |110000010}} | |
| |110000101}} | | |100001011}} | | |000010111}} | | |000101110}} | | |001011100}} | | |010111000}} | | |101110000}} | | |011100001}} | | |111000010}} | |
| |010000101}} | | |100001010}} | | |000010101}} | | |000101010}} | | |001010100}} | | |010101000}} | | |101010000}} | | |010100001}} | | |101000010}} | |
| |010000111}} | | |100001110}} | | |000011101}} | | |000111010}} | | |001110100}} | | |011101000}} | | |111010000}} | | |110100001}} | | |101000011}} | |
| |010000011}} | | |100000110}} | | |000001101}} | | |000011010}} | | |000110100}} | | |001101000}} | | |011010000}} | | |110100000}} | | |101000001}} | |
| |010000001}} | | |100000010}} | | |000000101}} | | |000001010}} | | |000010100}} | | |000101000}} | | |001010000}} | | |010100000}} | | |101000000}} | |
| |000000001}} | | |000000010}} | | |000000100}} | | |000001000}} | | |000010000}} | | |000100000}} | | |001000000}} | | |010000000}} | | |100000000}} |
{|class=“wikitable” style="text-align:center; background:#FFFFFF; border-width:0;“|+ 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|
| | 2 | |
| | 6 | |
| | 7 | |
| | 5 | |
| | 5 | |
| | 8 | |
| | 3 | |
| | 4 | |
| | 3 | |
| | 6 | |
| | 6 | |
| | 4 | |
| | 17 | |
| | 9 | |
| | 2 | |
| | 19 | |
| | 5 | |
| | 7 | |
| | 4 | Two-dimensional Gray codeFile:16QAM Gray Coded.svg|200px|thumb|right|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-CodeIf 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 ↔ {{mono|00}}, 1 ↔ {{mono|01}}, 2 ↔ {{mono|11}}, 3 ↔ {{mono|10}} } establishes an isometry between the metric space over the finite field mathbb{Z}_2^2 with the metric given by the Hamming distance and the metric space over the finite ring 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{Z}_2^{2m} and 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{Z}_2^2 of ring-linear codes from mathbb{Z}_4. Related codes {{Excessive citations|section|date=March 2021|details=Too many references makes the text hard to read}}There are a number of binary codes similar to Gray codes, including:
- {{anchor|Datex}}Datex codes aka Giannini codes (1954), as described by Carl P. Spaulding, use a variant of O’Brien code II.
- {{anchor|Varec}} Codes used by Varec (ca. 1954), use a variant of O’Brien code I as well as base-12 and base-16 Gray code variants.
- {{anchor|Lucal|MRB}}Lucal code (1959) aka modified reflected binary code (MRB)
- {{anchor|Gillham}}Gillham code (1961/1962), uses a variant of Datex code and O’Brien code II.
- {{anchor|Leslie-Russell}}Leslie and Russell code (1964)
- {{anchor|RRE}}Royal Radar Establishment code
- {{anchor|Hoklas}}Hoklas code (1988)
The following binary-coded decimal (BCD) codes are Gray code variants as well:
- {{anchor|Petherick|RAE}}Petherick code (1953), also known as Royal Aircraft Establishment (RAE) code.
- {{anchor|Foss|O’Brien|O’Brien I|Watts|O’Brien II}}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.)
- {{anchor|Gray excess|Gray–Stibitz}}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.
- {{anchor|Tompkins|Tompkins I|Tompkins II}}Tompkins codes I and II (1956)
- {{anchor|Glixon}}Glixon code (1957), sometimes ambiguously also called modified Gray code
{| class=“wikitable” style="text-align:center;“|+ 4-bit unit-distance BCD codes|
| Name > | Bit > | 0 | style="background:lightgray;width:1em” | | 2 | style="background:lightgray;width:1em” | | 4 | style="background:lightgray;width:1em” | | 6 | style="background:lightgray;width:1em” | | 8 | style="background:lightgray;width:1em” | | Hamming weight>Weights | Tracks | Method of complements | Compl. > | Cyclic > | 5s >| Comment | |
| | |
| {{anchor | Gray BCD > | 4 | {{mono | | 0}} | {{mono | | 0}} | {{mono | | 0}} | {{mono | | 0}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | rowspan=“4” | | 4 (3) | rowspan=“4” | | (2, 4, 8, 16) | rowspan=“4” | | | |
| 3 | {{mono | | 0}} | {{mono | | 0}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | |
| 2 | {{mono | | 0}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | {{mono | | 0}} | {{mono | | 0}} | |
| 1 | {{mono | | {{mono | | {{mono | | 0}} | {{mono | | {{mono | | {{mono | | 0}} | {{mono | | {{mono|1}} | |
| | |
| {{anchor | Paul > | 4 | style="background:#0FF” | 1}} | {{mono | | 0}} | {{mono | | 0}} | {{mono | | 0}} | {{mono | | {{mono | | {{mono | | 1—3 | rowspan=“4” | | No | rowspan=“4” | | No | rowspan=“4“| | |
| 3 | {{mono | | 0}} | {{mono | | 0}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | |
| 2 | {{mono | | 0}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | {{mono | | 0}} | {{mono | | 0}} | |
| 1 | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | {{mono | | 0}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | {{mono | | 0}} | style="background:#0FF” | 1}} | |
| | |
| Glixon | style="background:lightgray” | 4 > | 0}} | {{mono | | 0}} | {{mono | | 0}} | {{mono | | 0}} | {{mono | | {{mono | | {{mono | | 0—3 | rowspan=“4” | | No | rowspan=“4” | | (shifted +1) | rowspan=“4“| | |
| 3 | {{mono | | 0}} | {{mono | | 0}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | {{mono|0}} | |
| 2 | {{mono | | 0}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | {{mono | | 0}} | {{mono | | 0}} | |
| 1 | {{mono | | {{mono | | {{mono | | 0}} | {{mono | | {{mono | | {{mono | | 0}} | {{mono | | 0}} | |
| | |
| Tompkins I | style="background:lightgray” | 4 > | 0}} | {{mono | | 0}} | {{mono | | 0}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | rowspan=“4” | | 2 | rowspan=“4” | | 2, 4, 10 | rowspan=“4” | | | |
| 3 | {{mono | | 0}} | {{mono | | 0}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | {{mono|0}} | |
| 2 | {{mono | | 0}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | {{mono | | 0}} | {{mono|0}} | |
| 1 | {{mono | | {{mono | | {{mono | | 0}} | {{mono | | 0}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | {{mono | | 0}} | |
| | |
| O’Brien I (Watts) | style="background:lightgray” | 4 > | 0}} | {{mono | | 0}} | {{mono | | 0}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | rowspan=“4” | | 4 | rowspan=“4” | | 2, 4, 10 | rowspan=“4” | | | |
| 3 | {{mono | | 0}} | {{mono | | 0}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | {{mono | | 0}} | {{mono | | 0}} | |
| 2 | {{mono | | 0}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | {{mono | | 0}} | |
| 1 | {{mono | | {{mono | | {{mono | | 0}} | {{mono | | 0}} | {{mono | | {{mono | | {{mono | | 0}} | |
| | |
| Petherick (RAE) | style="background:lightgray” | 4 > | 0}} | {{mono | | 0}} | {{mono | | 0}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | rowspan=“4” | | 3 | rowspan=“4” | | 2, 10 | rowspan=“4” | | | |
| 3 | style="background:#0FF” | 1}} | {{mono | | 0}} | {{mono | | {{mono | | {{mono | | 0}} | {{mono | | 0}} | style="background:#0FF” | 1}} | |
| 2 | {{mono | | 0}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | {{mono | | 0}} | |
| 1 | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | {{mono | | 0}} | {{mono | | 0}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | |
| | |
| O’Brien II | style="background:lightgray” | 4 > | 0}} | {{mono | | 0}} | {{mono | | 0}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | rowspan=“4” | | 3 | rowspan=“4” | | 2, 10 | rowspan=“4” | | | |
| 3 | {{mono | | 0}} | {{mono | | {{mono | | {{mono | | {{mono | | {{mono | | 0}} | {{mono | | 0}} | |
| 2 | {{mono | | {{mono | | {{mono | | {{mono | | 0}} | {{mono | | {{mono | | {{mono | | {{mono | | 0}} | |
| 1 | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | {{mono | | 0}} | {{mono | | 0}} | {{mono | | 0}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | |
| | |
| {{anchor | Susskind > | 4 | {{mono | | 0}} | {{mono | | 0}} | {{mono | | {{mono | | {{mono | | {{mono | | {{mono | | {{mono | | 1—4 | rowspan=“4” | | 9 | rowspan=“4” | | Yes | rowspan=“4“| | |
| 3 | {{mono | | 0}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | {{mono | | 0}} | |
| 2 | {{mono | | {{mono | | {{mono | | {{mono | | 0}} | {{mono | | {{mono | | {{mono | | {{mono | | 0}} | |
| 1 | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | {{mono | | 0}} | {{mono | | 0}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | |
| | |
| {{anchor | Klar > | 4 | {{mono | | 0}} | {{mono | | 0}} | {{mono | | {{mono | | {{mono | | {{mono | | {{mono | | {{mono | | 0—4 | rowspan=“4” | | 9 | rowspan=“4” | | Yes | rowspan=“4“| | |
| 3 | {{mono | | 0}} | {{mono | | {{mono | | {{mono | | {{mono | | {{mono | | 0}} | {{mono | | 0}} | |
| 2 | {{mono | | 0}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | {{mono | | 0}} | |
| 1 | {{mono | | {{mono | | {{mono | | {{mono | | 0}} | {{mono | | {{mono | | {{mono | | {{mono | | 0}} | |
| | |
| Tompkins II | style="background:lightgray” | 4 > | 0}} | {{mono | | 0}} | {{mono | | 0}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | rowspan=“4” | | 2 | rowspan=“4” | | 2, 10 | rowspan=“4” | | | |
| 3 | {{mono | | 0}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | {{mono | | 0}} | {{mono|0}} | |
| 2 | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | {{mono | | 0}} | {{mono | | 0}} | {{mono | | {{mono | | {{mono|1}} | |
| 1 | {{mono | | {{mono | | {{mono | | {{mono | | 0}} | {{mono | | {{mono | | {{mono | | {{mono | | 0}} | |
| | |
| {{nowrap | Excess-3 Gray}} > | 4 | {{mono | | 0}} | {{mono | | 0}} | {{mono | | {{mono | | {{mono | | {{mono | | {{mono | | {{mono | | 1—4 | rowspan=“4” | | 9 | rowspan=“4” | | Yes | rowspan=“4“| | |
| 3 | {{mono | | {{mono | | {{mono | | {{mono | | {{mono | | {{mono | | {{mono | | {{mono | | {{mono | | 0}} | |
| 2 | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | {{mono | | 0}} | {{mono | | 0}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | |
| 1 | {{mono | | 0}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | {{mono | | 0}} | style="background:#0FF” | 1}} | style="background:#0FF” | 1}} | {{mono | | 0}} |
See also
Notes{{reflist|group=“nb“|refs=Sequence 0, 1, 0, 2, 0, 1, 0, 3, … {{OEIS|id=A007814}}.By interchanging and inverting three bit rows, the O’Brien code II and the Petherick code can be transferred into each other.By swapping two pairs of bit rows, individually shifting four bit rows and inverting one of them, the Glixon code and the O’Brien code I can be transferred into each other.By applying a simple inversion rule, the Gray code and the O’Brien code I can be translated into the 8421 pure binary code and the 2421 Aiken code, respectively, to ease arithmetic operations.REF, Evans, 1961, C, There are several Gray code variants which are called “modified” of some sort: The Glixon code is sometimes called modified Gray code.REF, Powell, 1968, D, The Lucal code is also called modified reflected binary code (MRB).REF, Sellers, Jr., Hsiao, Bearnson, 1968, E, The O’Brien code I or Watts code is sometimes referred to as reflected binary modified Gray code.REF, Savard, 2018, F, For O’Brien codes I and II and Petherick, Susskind, Klar as well as Excess-3 Gray codes, a 9s complement can be derived by inverting the most-significant (fourth) binary digit.For Tompkins code II, a 9s complement can be derived by inverting the first three digits and swapping the two middle binary digits.For Gray BCD, Paul and Klar codes, the number of necessary reading tracks can be reduced from 4 to 3 if inversion of one of the middle tracks is acceptable.Depending on a code’s target application, the Hamming weights of a code can be important properties beyond coding-theoretical considerations also for physical reasons. Under some circumstances the all-cleared and/or all-set states must be omitted (f.e. to avoid non-conductive or short-circuit conditions), it may be desirable to keep the highest used weight as low as possible (f.e. to reduce power consumption of the reader circuit) or to keep the variance of used weights small (f.e. to reduce acoustic noise or current fluctuations).Other unit-distance BCD codes include the non-Gray code related 5-bit Libaw–Craig and the 1-2-1 code.}}References
(6 pages)BOOK, Otto Schäffler (1838-1928). Pionier des Telephons, der Telegraphie und der Lochkarte sowie Erbauer der ersten Wiener Telephonzentrale, de, en, Heinrich “Heinz” Josef, Zemanek, Heinrich Josef Zemanek, Blätter für Technikgeschichte, 41–43 (1979–1981), 0067-9127, 1, 81–118, Vienna, Austria, Technisches Museum für Industrie und Gewerbe, Forschungsinstitut für Technikgeschichte / Springer-Verlag, 1983-12-01, 3-21181779-4, 952698275, WEB, Binary counter, George Robert, Stibitz, George Robert Stibitz, {{US patent, 2307868, . Serial No. 420537 |orig-date=1941-11-26 |date=1943-01-12 |publisher=Bell Telephone Laboratories, Incorporated |publication-place=New York, USA |url=https://patents.google.com/patent/US2307868?oq=US2307868 |access-date=2020-05-24 |quote-page=2, right column, rows 43–73 |quote=[…] 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 […]}} patentimages.storage.googleapis.com/51/a5/54/f2f6faf23da3fb/US2307868.pdf" title="https:/-/web.archive.org/web/20201217181617patentimages.storage.googleapis.com/51/a5/54/f2f6faf23da3fb/US2307868.pdf">weblink (4 pages)BOOK, Frank, Gray, Frank Gray (researcher), Pulse Code Communication, 1953-03-17, 1947-11-13, Bell Telephone Laboratories, Incorporated, New York, USA, {{US patent, 2632058, . Serial No. 785697 |url=https://patentimages.storage.googleapis.com/a3/d7/f2/0343f5f2c0cf50/US2632058.pdf |access-date=2020-08-05 |url-status=live |archive-url=https://web.archive.org/web/20200805094312patentimages.storage.googleapis.com/a3/d7/f2/0343f5f2c0cf50/US2632058.pdf |archive-date=2020-08-05}} (13 pages)BOOK, Jack, Breckman, Encoding Circuit, 1956-01-31, 1953-12-31, US Secretary of the Army, Long Branch, New Jersey, USA, {{US patent, 2733432, . Serial No. 401738 |url=https://patentimages.storage.googleapis.com/46/90/d6/2ed9fb7a77e643/US2733432.pdf |access-date=2020-08-05 |url-status=live |archive-url=https://web.archive.org/web/20200805094857patentimages.storage.googleapis.com/46/90/d6/2ed9fb7a77e643/US2733432.pdf |archive-date=2020-08-05}} (8 pages)BOOK, Earl Albert, Ragland, Harry B., Schultheis, Jr., Direction-Sensitive Binary Code Position Control System, 1958-02-11, 1953-10-16, Bendix Aviation Corporation, North Hollywood, California, USA, {{US patent, 2823345, . Serial No. 386524 |url=https://patentimages.storage.googleapis.com/ad/74/b1/2b0aa965d0c1df/US2823345.pdf |access-date=2020-08-05 |url-status=live |archive-url=https://web.archive.org/web/20200805095445patentimages.storage.googleapis.com/ad/74/b1/2b0aa965d0c1df/US2823345.pdf |archive-date=2020-08-05}} (10 pages)BOOK, Sol, Domeshek, Stewart, Reiner, Automatic Rectification System, 1958-06-24, 1954-01-08, US Secretary of the Navy, {{US patent, 2839974, . Serial No. 403085 |url=https://patentimages.storage.googleapis.com/9d/bf/65/e676a661e1217e/US2839974.pdf |access-date=2020-08-05 |url-status=live |archive-url=https://web.archive.org/web/20200805100327patentimages.storage.googleapis.com/9d/bf/65/e676a661e1217e/US2839974.pdf |archive-date=2020-08-05}} (8 pages)BOOK, Reflected binary code counter, George D., Hulst, International Telephone and Telegraph Corporation (ITT), Nutley, New Jersey, USA, 1962-02-06, 1957-11-15, {{US patent, 3020481, . Serial No. 696793 |url=https://patentimages.storage.googleapis.com/a2/e5/21/f5abbb073bf21c/US3020481.pdf |access-date=2020-08-06 |url-status=live |archive-url=https://web.archive.org/web/20200806210248patentimages.storage.googleapis.com/a2/e5/21/f5abbb073bf21c/US3020481.pdf |archive-date=2020-08-06}} (5 pages)BOOK, The Gray Code, CDMTCS Research Report Series, Robert “Bob” William, Doran, Robert William Doran, Centre for Discrete Mathematics and Theoretical Computer Science, University of Auckland, New Zealand, March 2007, CDMTCS-304,www.cs.auckland.ac.nz/research/groups/CDMTCresearchreports/304bob.pdf, 2020-05-23, live, https:web.archive.org/web/20200522091749www.cs.auckland.ac.nz/research/groups/CDMTCS//researchreports/304bob.pdf, 2020-05-22, (25 pages)WEB, Pulse-Count Coder, Frank P., Turvey, Jr., {{US patent, 2845617A, . Serial No. 585494 |date=1958-07-29 |orig-date=1956-05-17 |publisher=International Telephone and Telegraph Corporation |location=Nutley, New Jersey, USA |url=https://patentimages.storage.googleapis.com/14/07/c6/404dafe27dc829/US2845617.pdf |access-date=2020-05-23 |url-status=live |archive-url=https://web.archive.org/web/20200523115811patentimages.storage.googleapis.com/14/07/c6/404dafe27dc829/US2845617.pdf |archive-date=2020-05-23}} (5 pages)JOURNAL, Shaft Angle Encoders Afford High Accuracy, C. Farrell, Winder, Electronic Industries (journal), Electronic Industries, Chilton Company, 18, 10, October 1959, 76–80,worldradiohistory.com/Archive-Tele-Tech/50s/Electronic-Industries-1959-10.pdf, 2018-01-14,web.archive.org/web/20200928132232/https://worldradiohistory.com/Archive-Tele-Tech/50s/Electronic-Industries-1959-10.pdf, 2020-09-28, 78, […] The type of code wheel most popular in optical encoders contains a cyclic binary code pattern designed to give a cyclic sequence of “on-off” 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 pulse-code modulation systems by Frank Gray (researcher), 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. […], JOURNAL, Karnaugh, Maurice, Maurice Karnaugh, The Map Method for Synthesis of Combinational Logic Circuits, (Transactions of the American Institute of Electrical Engineers, Part I: Communication and Electronics), 72, 5, 593–599, November 1953, 1953-04-23, 1953-03-17, Paper 53-217, 10.1109/TCE.1953.6371932, 51636736,philectrosophy.com/documents/The%20Map%20Method%20For%20Synthesis%20of.pdf, 2017-04-16,philectrosophy.com/documents/The%20Map%20Method%20For%20Synthesis%20of.pdf," title="web.archive.org/web/20170416232229philectrosophy.com/documents/The%20Map%20Method%20For%20Synthesis%20of.pdf,">web.archive.org/web/20170416232229philectrosophy.com/documents/The%20Map%20Method%20For%20Synthesis%20of.pdf, 2017-04-16, (NB. Also contains a short review by Samuel H. Caldwell.)BOOK, Digital Design: Principles & Practices, Wakerly, John F., 1994, Prentice Hall, New Jersey, USA, 0-13-211459-3, 48–49, 222, (NB. The two page sections taken together say that K-maps are labeled with Gray code. The first section says that they are labeled with a code that changes only one bit between entries and the second section says that such a code is called Gray code.)BOOK, Boolean Reasoning – The Logic of Boolean Equations, 3.9.2 Maps, Frank Markham, Brown, reissue of 2nd, Dover Publications, Inc., Mineola, New York, USA, 2012, 2003, 1990, 978-0-486-42785-0, 49, 49, […] Karnaugh’s map orders the arguments of the discriminants according to the reflected binary code, also called the Gray code. […], (xii+291+3 pages) www2.fiit.stuba.sk/~kvasnicka/Free%20books/Brown_Boolean%20Reasoning.pdf" title="web.archive.org/web/20170416231752www2.fiit.stuba.sk/~kvasnicka/Free%20books/Brown_Boolean%20Reasoning.pdf">1st editionBOOK, Ein Minimisierungsverfahren zur Synthese von Schaltkreisen (Minimisierungsgraphen), de, Wolfgang, Händler, Wolfgang Händler, Technische Hochschule Darmstadt, Potsdam, Germany, 1958, D 17, Dissertation,books.google.com/books?id=D58TAQAAIAAJ, (73 pages+app.) weblinkBOOK, Taschenbuch der Nachrichtenverarbeitung, de, Karl W., Steinbuch, Karl W. Steinbuch, Siegfried W., Wagner, Erich R., Berger, Wolfgang, Händler, Wolfgang Händler, 1967, 1962, 2, Springer-Verlag OHG, Berlin, Germany, Title No. 1036, 67-21079, 64, 1034–1035, 1036, 1038, 64, […] Ãœbersichtlich ist die Darstellung nach Wolfgang Händler, Händler, die sämtliche Punkte, numeriert nach dem Gray-Code […], 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., WEB, Informatik Sammlung Erlangen (ISER), 2012-03-13, Friedrich-Alexander Universität, Erlangen, Germany, de,www.rrze.fau.de/wir-ueber-uns/kooperationen/iser.shtml, 2017-04-12,web.archive.org/web/20170516154655/https://www.rrze.fau.de/wir-ueber-uns/kooperationen/iser.shtml, 2017-05-16, WEB, Informatik Sammlung Erlangen (ISER) – Impressum, 2012-03-13, Friedrich-Alexander Universität, Erlangen, Germany, de,www.iser.uni-erlangen.de:80/index.php?ort_id=327&tree=0, 2017-04-15, live,www.iser.uni-erlangen.de/index.php?ort_id=327&tree=0," title="web.archive.org/web/20120226004316www.iser.uni-erlangen.de/index.php?ort_id=327&tree=0,">web.archive.org/web/20120226004316www.iser.uni-erlangen.de/index.php?ort_id=327&tree=0, 2012-02-26, JOURNAL, Kautz, William H., William H. Kautz, Unit-Distance Error-Checking Codes, IRE Transactions on Electronic Computers, 0367-9950, EC-7, 2, June 1958, 10.1109/TEC.1958.5222529, 26649532, 179–180, (2 pages)BOOK, Richard Kohler, Richards, Digital Design, Snake-in-the-Box Codes, Wiley-Interscience, John Wiley & Sons, Inc., Ames, Iowa, USA, New York, USA, January 1971, 0-471-71945-5, 73-147235, 206–207, (12+577+1 pages)BOOK, Frederick F., Sellers, Jr., Mu-Yue, Hsiao, Leroy W., Bearnson, Error Detecting Logic for Digital Computers, McGraw-Hill Book Company, New York, USA, 1st, 439460, 68-16491, 152–164, November 1968, JOURNAL, Harold M., Lucal, Arithmetic Operations for Digital Computers Using a Modified Reflected Binary, IRE Transactions on Electronic Computers, EC-8, 4, 449–458, December 1959, 0367-9950, 10.1109/TEC.1959.5222057, 206673385,ieeexplore.ieee.org/document/5222057, (10 pages)WEB, Digital coding and translating system, Carl P., Spaulding, Datex Corporation, Monrovia, California, USA, 1965a, 1965-01-12, 1954-03-09, {{US patent, 3165731A, . Serial No. 415058 |url=https://patentimages.storage.googleapis.com/7f/1d/09/6a9b1fa3e67cb8/US3165731.pdf |access-date=2018-01-21 |url-status=live |archive-url=https://web.archive.org/web/20200805101618patentimages.storage.googleapis.com/7f/1d/09/6a9b1fa3e67cb8/US3165731.pdf |archive-date=2020-08-05}} (28 pages)BOOK, How to Use Shaft Encoders, Carl P., Spaulding, 1965b, 1965-07-12, Datex Corporation, Monrovia, California, USA, (85 pages)WEB, Control Systems, Frederic A., Foss, 1960-12-27, 1954-12-17, {{US patent, 2966670A, . Serial No. 475945 |publisher=International Business Machines Corp |pages=Fig. 7, Fig. 8, Fig. 11 |no-pp=yes |url=https://patentimages.storage.googleapis.com/3d/a8/16/1dc616c432ca95/US2966670.pdf |access-date=2020-08-05 |url-status=live |archive-url=https://web.archive.org/web/20200621145238patentimages.storage.googleapis.com/3d/a8/16/1dc616c432ca95/US2966670.pdf |archive-date=2020-06-21}} (14 pages) (NB. The author called his code 2*-4-2-1 (+9-±7-±3-±1) reflected decimal code.)JOURNAL, The Use of a Reflected Code in Digital Control Systems, Frederic A., Foss, December 1954, IRE Transactions on Electronic Computers, EC-3, 4, 2168-1740, 10.1109/IREPGELC.1954.6499244, 1–6, (6 pages)JOURNAL, Joseph A., O’Brien, Cyclic Decimal Codes for Analogue to Digital Converters, (Transactions of the American Institute of Electrical Engineers, Part I: Communication and Electronics), Bell Telephone Laboratories, Whippany, New Jersey, USA, 75, 2, May 1956, 1955-11-15, 1955-06-23, 0097-2452, 10.1109/TCE.1956.6372498, Paper 56-21, 51657314, 120–122,pdfslide.net/documents/cyclic-decimal-codes-for-analogue-to-digital-converters.html, 2020-05-18, (3 pages) (NB. This paper was prepared for presentation at the AIEE Winter General Meeting, New York, USA, 1956-01-30 to 1956-02-03.)JOURNAL, Howard E., Tompkins, Unit-Distance Binary-Decimal Codes for Two-Track Commutation, September 1956, 1956-07-16, IRE Transactions on Electronic Computers, 0367-9950, EC-5, 3, 10.1109/TEC.1956.5219934, Correspondence, Moore School of Electrical Engineering, University of Pennsylvania, Philadelphia, Pennsylvania, USA, 139,dokumen.tips/documents/unit-distance-binary-decimal-codes-for-two-track-commutation.html, 2020-05-18, (1 page)JOURNAL, March 1957, Can You Take Advantage of the Cyclic Binary-Decimal Code?, Harry Robert, Glixon, Control Engineering (magazine), Control Engineering, 0010-8049, Technical Publishing Company, a division of Dun-Donnelley Publishing Corporation, Dun & Bradstreet Corp., 4, 3, 87–91,books.google.com/books?id=-_5IAQAAIAAJ, (5 pages)BOOK, Taschenbuch der Nachrichtenverarbeitung, de, Karl W., Steinbuch, Karl W. Steinbuch, 1962, 1, Springer-Verlag OHG, Karlsruhe, Germany, Berlin / Göttingen / New York, 62-14511, 71–74, 97, 761–764, 770, 1080–1081, BOOK, Taschenbuch der Informatik – Band II – Struktur und Programmierung von EDV-Systemen, de, Karl W., Steinbuch, Karl W. Steinbuch, Wolfgang, Weber, Traute, Heinemann, 1974, 1967, 3, 2, Taschenbuch der Nachrichtenverarbeitung, Springer Verlag, Berlin, Germany, 3-540-06241-6, 73-80607, 98–100, BOOK, Digital Electronics, Folkert, Dokter, Jürgen, Steinhauer, 2.4. Coding numbers in the binary system, 1973-06-18, Philips Technical Library (PTL) / Macmillan Education, The Macmillan Press Ltd. / N. V. Philips’ Gloeilampenfabrieken, Reprint of 1st English, Eindhoven, Netherlands, 333-13360-9, 978-1-349-01419-4, 10.1007/978-1-349-01417-0, 32, 39, 50–53,books.google.com/books?id=hlRdDwAAQBAJ, 2020-05-11, 53, […] The #Datex, Datex code […] uses the #O’Brien II, 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. […], {{Dead link|date=July 2023 |bot=InternetArchiveBot |fix-attempted=yes }} (270 pages)BOOK, Folkert, Dokter, Jürgen, Steinhauer, Digitale Elektronik in der Meßtechnik und Datenverarbeitung: Theoretische Grundlagen und Schaltungstechnik, 2.4.4.6. Einschrittige Kodes, de, Philips Fachbücher, Deutsche Philips GmbH, Hamburg, Germany, I, 1975, 1969, improved and extended 5th, 3-87145-272-6, 41, 48, 51, 58, 60–61, (xii+327+3 pages)BOOK, Edward John, Petherick, A Cyclic Progressive Binary-coded-decimal System of Representing Numbers, October 1953, Technical Note MS15, Royal Aircraft Establishment (RAE), Farnborough, UK, (4 pages) (NB. Sometimes referred to as A Cyclic-Coded Binary-Coded-Decimal System of Representing Numbers.)BOOK, Edward John, Petherick, A. J., Hopkins, Some Recently Developed Digital Devices for Encoding the Rotations of Shafts, 1958, Technical Note MS21, Royal Aircraft Establishment (RAE), Farnborough, UK, PERIODICAL, Digitizer als Analog-Digital-Wandler in der Steuer-, Meß- und Regeltechnik, Technische Mitteilungen, Relais, elektronische Geräte, Steuerungen, de, May 1963, 13, Franz Baumgartner (FraBa), Cologne-Niehl, Germany, 1–2,www.posital.com/media/posital_media/documents/TechnischeMitteilung_May1963.pdf, 2020-05-21,web.archive.org/web/20200521211656/https://www.posital.com/media/posital_media/documents/TechnischeMitteilung_May1963.pdf, 2020-05-21, 1–2, […] 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 Ãœbergabe 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 #Petherick, Petherick-Code. […], (4 pages)JOURNAL, The Design of a Pneumatic Position Encoder, C. J., Charnley, R. E., Bidgood, G. E. T., Boardman, IFAC Proceedings Volumes, The College of Aeronautics, Cranfield, Bedford, England, 2, 3, October 1965, 75–88, Chapter 1.5., 10.1016/S1474-6670(17)68955-9,ac.els-cdn.com/S1474667017689559/1-s2.0-S1474667017689559-main.pdf?_tid=f0c1e48e-f95b-11e7-ad9a-00000aab0f01&acdnat=1515956073_9006e89e176c6a840b5454c38525240b, 2018-01-14, {{Dead link|date=October 2023 |bot=InternetArchiveBot |fix-attempted=yes }}BOOK, Analog to digital encoder, Edwin L., Wheeler, Conrac Corporation, New York, USA, 1969-12-30, 1968-04-05, {{US patent, 3487460A, . Serial No. 719026 (397812) |url=https://patentimages.storage.googleapis.com/f0/c0/60/9c3231f7e8ed44/US3487460.pdf |access-date=2018-01-21 |url-status=live |archive-url=https://web.archive.org/web/20200805102804patentimages.storage.googleapis.com/f0/c0/60/9c3231f7e8ed44/US3487460.pdf |archive-date=2020-08-05 |quote-page=5, left column 9, rows 15–22 |quote=[…] The MOA-GILLHAM 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 REF, Spaulding, 1965a, 3,165,731, plain, . 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)WEB, Telemetering and supervisory control system having normally continuous telemetering signals, US3397386A, Bernard W., Bishup, Anthony A., Repeta, Frank C., Giarrizzo, Leeds & Northrup, Leeds and Northrup Co., 1968-08-13, 1963-04-03,patents.google.com/patent/US3397386A/en, weblinkJOURNAL, Carla Diane, Savage, Carla Diane Savage, Long cycles in the middle two levels of the Boolean lattice, Ars Combinatoria (journal), Ars Combinatoria, 0381-7032, 35, A, 1997-01-16, North Carolina State University, Raleigh, North Carolina, USA, 15975960, 10.1.1.39.2249, 97–108,citeseerx.ist.psu.edu/viewdoc/download;jsessionid=DACA73A3D791549CE05443B7948527EC?doi=10.1.1.39.2249&rep=rep1&type=pdf, 2020-05-13, live,citeseerx.ist.psu.edu/viewdoc/download;jsessionid=DACA73A3D791549CE05443B7948527EC?doi=10.1.1.39.2249&rep=rep1&type=pdf," title="web.archive.org/web/20200513191647citeseerx.ist.psu.edu/viewdoc/download;jsessionid=DACA73A3D791549CE05443B7948527EC?doi=10.1.1.39.2249&rep=rep1&type=pdf,">web.archive.org/web/20200513191647citeseerx.ist.psu.edu/viewdoc/download;jsessionid=DACA73A3D791549CE05443B7948527EC?doi=10.1.1.39.2249&rep=rep1&type=pdf, 2020-05-13, (15 pages)WEB, Altitude – MODEC ASCII, Darryl, Phillips, 2012-07-26, 1998, AirSport Avionics,www.airsport-corp.com/modecascii.txt,www.airsport-corp.com/modecascii.txt," title="web.archive.org/web/20120726003224www.airsport-corp.com/modecascii.txt,">web.archive.org/web/20120726003224www.airsport-corp.com/modecascii.txt, 2012-07-26, WEB, Aviation Gray Code: Gillham Code Explained, 2010-12-03, K., Stewart, Custom Computer Services (CCS),www.ccsinfo.com/forum/viewtopic.php?p=140960#140960, 2018-01-14, live,www.ccsinfo.com/forum/viewtopic.php?p=140960," title="web.archive.org/web/20180116184525www.ccsinfo.com/forum/viewtopic.php?p=140960,">web.archive.org/web/20180116184525www.ccsinfo.com/forum/viewtopic.php?p=140960, 2018-01-16, BOOK, A cyclic progressive decimal code for simple translation to decimal and analogue outputs, William “Bill” H. P., Leslie, A., Russell, National Engineering Laboratory, East Kilbride, Glasgow, UK, 1964, Report, NEL Report 129, (17 pages)JOURNAL, Some Binary Codes and a Novel Five-Channel Code, A., Russell, August 1964, Control (Systems, Instrumentation, Data Processing, Automation, Management, incorporating Automation Progress), 8, 74, Special Features, Morgan-Grampain (Publishers) Limited, London, UK, 399–404,books.google.com/books?id=jf4eAAAAMAAJ, 2020-06-22, (6 pages)BOOK, Proceedings of the Fourteenth International Machine Tool Design and Research Conference, 12–14 September 1973, The work on NC at NEL, William “Bill” H. P., Leslie, Franz, Koenigsberger, Stephen Albert, Tobias, The Macmillan Press Ltd, 1974, 333-14913-0, 978-1-34901921-2, 73-16545, 215–224 [215, 217], 10.1007/978-1-349-01921-2_30,books.google.com/books?id=bFVdDwAAQBAJ&pg=PA215,link.springer.com/chapter/10.1007/978-1-349-01921-2_30, 2020-05-21, 2022-04-07,web.archive.org/web/20220407160744/https://books.google.com/books?id=bFVdDwAAQBAJ&pg=PA215, dead, BOOK, Eric Jeffrey, Wightman, Instrumentation in Process Control, 1972, 1, Butterworth & Co (Publishers) Ltd, London, UK, 0-408-70293-1, Chapter 6. Displacement measurement, 122–123,books.google.com/books?id=8WEhBQAAQBAJ&pg=PA122,books.google.com/books?id=8WEhBQAAQBAJ, 122–123, […] 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)JOURNAL, …accurate liquid level metering – at ANY DISTANCE!, Petroleum Refiner, 33, 9, 0096-6517, Gulf Publishing Company, September 1954, Advertisement, 368, 368, […] The complete dispatching operation, gauging, and remote control is integrated into one single unitized system when a “Varec” Pulse Code Telemetering System is installed. […],books.google.com/books?id=B2Q7AAAAMAAJ&q=%22Varec+pulse+code%22, BOOK, Installation and Operations Manual for the Model 1900 Micro 4-Wire Transmitter, Encoder Pulse Format, Whessoe Varec, Inc., Cypress, California, USA, 33-08461, January 1993, 1991-07-01, 04-4–04-8,www.varec.com/web/wp-content/uploads/2018/05/IOM016_1900.pdf, 2020-05-16, live,web.archive.org/web/20200516191438/https://www.varec.com/web/wp-content/uploads/2018/05/IOM016_1900.pdf, 2020-05-16, (38 pages) (NB. Position 5 for “Inches” on page 04-8 should read “0111” rather than “1111”.)BOOK, Varec Model 1900 – Micro 4-Wire Transmitter (BSAP to Mark / Space Protocol (MSP)) – Application Notes, 2.2.3.3 MSP Level Data Format, Emerson Electric, 11–14,www.emerson.com/documents/automation/manuals-guides-application-notes-varec-model-1900-protocol-ras-en-133360.pdf, 2020-05-16, live,web.archive.org/web/20200516183338/https://www.emerson.com/documents/automation/manuals-guides-application-notes-varec-model-1900-protocol-ras-en-133360.pdf, 2020-05-16, (vi+33 pages)WEB, Abtastvorrichtung zur digitalen Weg- oder Winkelmessung, de, Archibald, Hoklas, {{ill, VEB Schiffselektronik Johannes Warnke, de, |via={{ill|DEPATIS|de}} |date=1989-09-06 |orig-date=1988-04-29 |id=GDR Patent DD271603A1. WP H 03 M / 315 194 8 |url=https://depatisnet.dpma.de/DepatisNet/depatisnet/DD000000271603A1_all_pages.pdf?window=1&space=menu&content=download_doc_verify&action=download_doc&docid=DD000000271603A1&so=asc&sf=vn&firstdoc=0&struct=&Cl=2&Bi=1&Ab=1&De=2&Dr=5&Pts=&Pa=&We=&Sr=&Eam=&Cor=&Aa=&NrFaxPages=5&pdfpage=2&pdfmatrix=1x1&origin=pdf_window&verify_str=null |access-date=2018-01-18 |archive-url=https://web.archive.org/web/20180118015239depatisnet.dpma.de/DepatisNet/depatisnet/DD000000271603A1_all_pages.pdf?window=1&space=menu&content=download_doc_verify&action=download_doc&docid=DD000000271603A1&so=asc&sf=vn&firstdoc=0&struct=&Cl=2&Bi=1&Ab=1&De=2&Dr=5&Pts=&Pa=&We=&Sr=&Eam=&Cor=&Aa=&NrFaxPages=5&pdfpage=2&pdfmatrix=1x1&origin=pdf_window&verify_str=null |archive-date=2018-01-18}} weblink weblinkWEB, Gray code – Unit distance code, Archibald, Hoklas, 2005,www.ahok.de/en/hoklas-code.html, 2018-01-15, live,www.ahok.de/en/hoklas-code.html," title="web.archive.org/web/20180115012854www.ahok.de/en/hoklas-code.html,">web.archive.org/web/20180115012854www.ahok.de/en/hoklas-code.html, 2018-01-15, WEB, Gray-Kode – Einschrittiger Abtastkode, de, Archibald, Hoklas, 2005,www.ahok.de/dt/hoklas-code.html, 2018-01-15, live,www.ahok.de/dt/hoklas-code.html," title="web.archive.org/web/20180115012827www.ahok.de/dt/hoklas-code.html,">web.archive.org/web/20180115012827www.ahok.de/dt/hoklas-code.html, 2018-01-15, WEB, Electronic Diode Matrix Decoder Circuits, James J., Klinkowski, {{US patent, 3309695A, . Serial No. 353845 |publisher=Burroughs Corporation |location=Detroit, Michigan, USA |date=1967-03-14 |orig-date=1964-03-23 |url=https://patentimages.storage.googleapis.com/71/6a/f4/044b43b93097e7/US3309695.pdf |access-date=2020-05-23 |url-status=live |archive-url=https://web.archive.org/web/20200523144819patentimages.storage.googleapis.com/71/6a/f4/044b43b93097e7/US3309695.pdf |archive-date=2020-05-23}} (5 pages) patentimages.storage.googleapis.com/97/33/74/aec52610026e59/USRE26780.pdf" title="https:/-/web.archive.org/web/20200805103732patentimages.storage.googleapis.com/97/33/74/aec52610026e59/USRE26780.pdf">weblinkWEB, Binary-coded decimal signal converter, James J., Klinkowski, {{US patent, 3504363A, . Serial No. 603926 |publisher=Burroughs Corporation |location=Detroit, Michigan, USA |date=1970-03-31 |orig-date=1966-12-22 |url=https://patentimages.storage.googleapis.com/65/d6/45/f7837c143fa3da/US3504363.pdf |access-date=2020-05-23 |url-status=live |archive-url=https://web.archive.org/web/20200523145655patentimages.storage.googleapis.com/65/d6/45/f7837c143fa3da/US3504363.pdf |archive-date=2020-05-23}} (7 pages)JOURNAL, (([title unknown])), Electrical Design News, 0012-7515, Rogers Publishing Company, 1967, 12, weblinkweblinkJOURNAL, Some Problems Of Angular Rotational Digital Converters, Györgyi, Tóth-Zentai, 23, 3–4, Periodica Polytechnica Electrical Engineering, 1979-10-05, 265–270 [266], Department of Electronics Technology, Technical University, Budapest, Hungary,pp.bme.hu/ee/article/view/4838, 2020-05-23, weblink (6 pages) (NB. Shows a 6-digit Watts code.)BOOK, Knuth, Donald Ervin, Donald Ervin Knuth, Enumeration and Backtracking / Generating all n-tuples, The Art of Computer Programming, Volume 4A: Combinatorial Algorithms, Part 1, 4A, 1, The Art of Computer Programming, 2014-09-12, Addison-Wesley Professional, 978-0-13348885-2, 442–443,books.google.com/books?id=IkuEBAAAQBAJ, (912 pages)JOURNAL, Pioneers Of Binary Coding, F. G., Heath, Journal of the Institution of Electrical Engineers, Institution of Engineering and Technology (IET), Manchester College of Science and Technology, Faculty of Technology of the University of Manchester, Manchester, UK, 7, 81, September 1961, 10.1049/jiee-3.1961.0300, 539–541,ieeexplore.ieee.org/document/5324416, 2020-06-22, (3 pages)BOOK, Kenneth W., Cattermole, Principles of pulse code modulation, 1, Iliffe Books Ltd. / American Elsevier Publishing Company, Inc., 1969, London, UK / New York, USA, Harlow, Essex, UK, 444-19747-8, 978-0-444-19747-4, 78-80432, 245, 434, 245, […] 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 {{citeref, Heath, 1961, Heath, plain, 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)BOOK, Cogwheels of the Mind: The Story of Venn Diagrams, Anthony William Fairbank, Edwards, Anthony William Fairbank Edwards, Johns Hopkins University Press, 2004, 0-8018-7434-3, Baltimore, Maryland, USA, 48, 50,books.google.com/books?id=7_0Thy4V3JIC&pg=PA65, JOURNAL, William M., Goodall, Television by Pulse Code Modulation, Bell System Technical Journal, 30, 1, 33–49, January 1951, 10.1002/j.1538-7305.1951.tb01365.x, (NB. Presented orally before the I.R.E. National Convention, New York City, March 1949.)JOURNAL, Girish S., Bhat, Carla Diane, Savage, Carla Diane Savage, Balanced Gray Codes, Electronic Journal of Combinatorics, 1996, 3,www.combinatorics.org/ojs/index.php/eljc/article/view/v3i1r25/, 1, 10.37236/1249, free, WEB, Ryan, Donohue, Synchronization in Digital Logic Circuits, 2003,www.stanford.edu/class/ee183/handouts_spr2003/synchronization_pres.pdf, 2018-01-15, live,web.archive.org/web/20180115012747/https://web.stanford.edu/class/ee183/handouts_spr2003/synchronization_pres.pdf, 2018-01-15, BOOK, Mehta, Huzefa, Owens, Robert Michael, Irwin, Mary Jane “Janie”, Some issues in gray code addressing, Proceedings of the Sixth Great Lakes Symposium on VLSI, 1996-03-22, 1066-1395, 10.1109/GLSV.1996.497616,ieeexplore.ieee.org/document/497616, IEEE Computer Society, 978-0-8186-7502-7, 178–181, 52837310, REPORT, Low Power Architecture Design and Compilation Techniques for High-Performance Processors, Ching-Long, Su, Chi-Ying, Tsui, Alvin M., Despain, 1994, Advanced Computer Architecture Laboratory, ACAL-TR-94-01,www.scarpaz.com/2100-papers/Power%20Estimation/su94-low%20power%20architecture%20and%20compilation.pdf, 2020-12-17, live,www.scarpaz.com/2100-papers/Power%20Estimation/su94-low%20power%20architecture%20and%20compilation.pdf," title="web.archive.org/web/20200726170626www.scarpaz.com/2100-papers/Power%20Estimation/su94-low%20power%20architecture%20and%20compilation.pdf,">web.archive.org/web/20200726170626www.scarpaz.com/2100-papers/Power%20Estimation/su94-low%20power%20architecture%20and%20compilation.pdf, 2020-07-26, JOURNAL, Hui, Guo, Sri, Parameswaran, 10.1016/j.sysarc.2010.03.003, 56, 4–6, April–June 2010, Shifted Gray encoding to reduce instruction memory address bus switching for low-power embedded systems, Journal of Systems Architecture, 180–190, WEB, Henry Gordon “Hank”, Dietz, The Aggregate Magic Algorithms: Gray Code Conversion, The Aggregate, Electrical and Computer Engineering Department, College of Engineering, University of Kentucky, 2002,aggregate.org/MAGIC/#Gray%20Code%20Conversion, 2020-12-16, live,aggregate.org/MAGIC/#Gray%2520Code%2520Conversion," title="web.archive.org/web/20201216122620aggregate.org/MAGIC/#Gray%2520Code%2520Conversion,">web.archive.org/web/20201216122620aggregate.org/MAGIC/#Gray%2520Code%2520Conversion, 2020-12-16, JOURNAL, Generalized Gray Codes with Applications, Guan, Dah-Jyh, Proceedings of the National Scientific Council, Republic of China, Part A, 22, 1998, 841–848, 10.1.1.119.1344, JOURNAL, I. Nengah, Suparta, A simple proof for the existence of exponentially balanced Gray codes, Electronic Journal of Combinatorics, 2005, 12, free, 10.37236/1986, JOURNAL, Mary Elizabeth, Flahive, Mary Elizabeth Flahive, Bella, Bose, Balancing cyclic R-ary Gray codes, Electronic Journal of Combinatorics, 2007, 14, free, 10.37236/949, JOURNAL, Raoul, Strackx, Frank, Piessens, Ariadne: A Minimal Approach to State Continuity, Usenix Security, 2016, 25,distrinet.cs.kuleuven.be/software/sce/ariadne.html, JOURNAL, Carla Diane, Savage, Carla Diane Savage, Peter, Winkler, Peter Winkler, Monotone Gray codes and the middle levels problem, Journal of Combinatorial Theory, Series A, 1995, 70, 0097-3165, 230–248, 2, 10.1016/0097-3165(95)90091-8, free, WEB, MATH 343 Applied Discrete Math Supplementary Materials, Goddyn, Luis, 1999, Department of Mathematics, Simon Fraser University,www.math.sfu.ca/~goddyn/Courses/343/supMaterials.pdf,people.math.sfu.ca/~goddyn/Courses/343/supMaterials.pdf," title="web.archive.org/web/20150217160033people.math.sfu.ca/~goddyn/Courses/343/supMaterials.pdf,">web.archive.org/web/20150217160033people.math.sfu.ca/~goddyn/Courses/343/supMaterials.pdf, 2015-02-17, JOURNAL, Joseph “Joe”, Sawada, Dennis Chi-Him, Wong, A Fast Algorithm to generate Beckett–Gray codes, Electronic Notes in Discrete Mathematics, 29, 571–577, 2007, 10.1016/j.endm.2007.07.091, PATENT, Spedding, Norman Bruce, 1994-10-28, A position encoder, NZ, 264738, {{failed verification|date=July 2015}}WEB, The following is a copy of the provisional patent filed on behalf of Industrial Research Limited on 1994-10-28 – NZ Patent 264738, Norman Bruce, Spedding, Industrial Research Limited, 1994-10-28, NZ Patent 264738,www.winzurf.co.nz/Single_Track_Grey_Code_Patent/Single_track_Grey_code_encoder_patent.pdf, 2018-01-14, live,www.winzurf.co.nz/Single_Track_Grey_Code_Patent/Single_track_Grey_code_encoder_patent.pdf," title="web.archive.org/web/20171029205005www.winzurf.co.nz/Single_Track_Grey_Code_Patent/Single_track_Grey_code_encoder_patent.pdf,">web.archive.org/web/20171029205005www.winzurf.co.nz/Single_Track_Grey_Code_Patent/Single_track_Grey_code_encoder_patent.pdf, 2017-10-29, JOURNAL, Single-Track Gray Codes, Hiltgen, Alain P., Kenneth G., Paterson, Marco, Brandestini, IEEE Transactions on Information Theory, 42, 5, September 1996, 1555–1561, 10.1109/18.532900, 857.94007,ieeexplore.ieee.org/iel1/18/11236/00532900.pdf, JOURNAL, Single-Track Circuit Codes, Hiltgen, Alain P., Kenneth G., Paterson, IEEE Transactions on Information Theory, 47, 6, September 2001, 2587–2595, 10.1109/18.945274, 10.1.1.10.8218,www.hpl.hp.com/techreports/2000/HPL-2000-81.pdf, 2018-01-15, live,www.hpl.hp.com/techreports/2000/HPL-2000-81.pdf," title="web.archive.org/web/20180115013155www.hpl.hp.com/techreports/2000/HPL-2000-81.pdf,">web.archive.org/web/20180115013155www.hpl.hp.com/techreports/2000/HPL-2000-81.pdf, 2018-01-15, JOURNAL, The Structure of Single-Track Gray Codes, Etzion, Tuvi, Moshe, Schwartz, IEEE Transactions on Information Theory, IT-45, 7, November 1999, 1998-05-17, 2383–2396, 10.1109/18.796379, 10.1.1.14.8333,etzion.net.technion.ac.il/files/2016/02/P54.pdf, 2018-01-15, live,etzion.net.technion.ac.il/files/2016/02/P54.pdf," title="web.archive.org/web/20180115022531etzion.net.technion.ac.il/files/2016/02/P54.pdf,">web.archive.org/web/20180115022531etzion.net.technion.ac.il/files/2016/02/P54.pdf, 2018-01-15, Technical Report CS0937 {{Webarchive|url=https://web.archive.org/web/20181215171438www.cs.technion.ac.il/users/wwwb/cgi-bin/tr-info.cgi?1998/CS/CS0937 |date=2018-12-15 }}WEB, Torsten, Sillke, 1997, 1993-03-01, Gray-Codes with few tracks (a question of Marco Brandestini),www.mathematik.uni-bielefeld.de/~sillke/PROBLEMS/gray, 2017-10-29, live,web.archive.org/web/20171029202303/https://www.math.uni-bielefeld.de/~sillke/PROBLEMS/gray, 2017-10-29, JOURNAL, Near Optimal Single-Track Gray Codes, Tuvi, Etzion, Kenneth G., Paterson, IEEE Transactions on Information Theory, IT-42, 3, 779–789, May 1996, 10.1109/18.490544, 10.1.1.14.1527,etzion.net.technion.ac.il/files/2016/02/P36.pdf, 2018-04-08, live,etzion.net.technion.ac.il/files/2016/02/P36.pdf," title="web.archive.org/web/20161030214251etzion.net.technion.ac.il/files/2016/02/P36.pdf,">web.archive.org/web/20161030214251etzion.net.technion.ac.il/files/2016/02/P36.pdf, 2016-10-30, JOURNAL, A Survey of Venn Diagrams: Symmetric Diagrams,www.combinatorics.org/Surveys/ds5/VennSymmEJC.html, Electronic Journal of Combinatorics, 2005-06-18, Frank, Ruskey, Frank Ruskey, Mark, Weston, 10.37236/26, free, Dynamic Surveys, BOOK, Alciatore, David G., Michael B., Histand, Mechatronics, 1999, McGraw–Hill Education – Europe, 978-0-07-131444-2,mechatronics.colostate.edu/, WEB, Krishna, Gray code for QAM, 2008-05-11,www.dsprelated.com/showthread/comp.dsp/96917-1.php, 2017-10-29, live,web.archive.org/web/20171029192539/https://www.dsprelated.com/showthread/comp.dsp/96917-1.php, 2017-10-29, BOOK, Massimiliano, Sala, Teo, Mora, Ludovic, Perret, Shojiro, Sakata, Carlo, Traverso, Gröbner Bases, Coding, and Cryptography,archive.org/details/grbnerbasescodin00sala, limited, 2009, Springer Science & Business Media, 978-3-540-93806-4, An Introduction to Ring-Linear Coding Theory, Marcus, Greferath, 220, BOOK, Kerdock and Preparata codes, Patrick, Solé, Encyclopedia of Mathematics, Michiel, Hazewinkel, Michiel Hazewinkel, Springer Science+Business Media, 2016, 978-1-4020-0609-8,www.encyclopediaofmath.org/index.php/Kerdock_and_Preparata_codes, live,web.archive.org/web/20171029191032/https://www.encyclopediaofmath.org/index.php/Kerdock_and_Preparata_codes, 2017-10-29, JOURNAL, (([title unknown])), David Silvester, Evans, 10–12, 87, Transactions, Institute of Measurement and Control, 1958,books.google.com/books?id=mwJvieqKGFsC&q=%22Watts+code%22, (NB. The Watts code was called W.R.D. code or Watts Reflected Decimal to distinguish it from other codes used at Hilger & Watts Ltd.)BOOK, Fundamentals of Digital Instrumentation, David Silvester, Evans, Hilger & Watts Ltd, London, UK, 1, 1960,books.google.com/books?id=gpVNAAAAYAAJ, 2020-05-24, (39 pages)BOOK, Digital Data: Their derivation and reduction for analysis and process control, Chapter Three: Direct Reading from Coded Scales, David Silvester, Evans, 1, Hilger & Watts Ltd / Interscience Publishers, London, UK, March 1961, 18–23,books.google.com/books?id=WOIJAAAAMAAJ, 2020-05-24, 20–23, […] Decoding. […] To decode C.P.B. or #Watts, 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 code, 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 line-by-line inversion rule on the W.R.D. code produces [a] pattern [of Aiken code, 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 high-speed 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”.)BOOK, Tables for the Binary Encoding of Angles, P. A., Newson, United Kingdom Atomic Energy Authority, Research Group, Atomic Energy Research Establishment, Harwell, UK, 1965, 1, H. M. Stationery Office,books.google.com/books?id=aUEgAQAAIAAJ, 2020-05-24, (12 pages)BOOK, Measurement of Neutron Spectra by Semi-Automatic Scanning of Recoil Protons in Photographic Emulsions, 3.2.2 Electromechanical Digitizers, P. W., Benjamin, G. S., Nicholls, United Kingdom Atomic Energy Authority, Atomic Weapons Research Establishment, UK, AWRE Report No. NR 5/63, U.S. Department of Energy, 1963, 8–10, 19,books.google.com/books?id=tD5YAAAAYAAJ&pg=PT4, books.googleusercontent.com/books/content?req=AKW5QadxAC8d1K6TXnjLMOKRt0wxLI2XEnyvs5b7wXlxCaZ4XdYW6qcmagghi7O1TLA4DL5nybOmiY0ue2ljEH9L2NvnP6Z6vN54dNvpxHnCIK79h1xoamtO5-waq-mDh2BPHJE9eJkmkoUl7uyPvrbWd7gNM8gpQKbx82WI_3gYiZHy-aHdQGQAFOkacvI1Kf52lu4i8_cuMNYztfa--t2eGiIk5ku1bP7T7n378Q2rNZRCMz3JCXeT0EzYNMkJzlpYSiP2FiQ8" title="https:/-/web.archive.org/web/20200525193325books.googleusercontent.com/books/content?req=AKW5QadxAC8d1K6TXnjLMOKRt0wxLI2XEnyvs5b7wXlxCaZ4XdYW6qcmagghi7O1TLA4DL5nybOmiY0ue2ljEH9L2NvnP6Z6vN54dNvpxHnCIK79h1xoamtO5-waq-mDh2BPHJE9eJkmkoUl7uyPvrbWd7gNM8gpQKbx82WI_3gYiZHy-aHdQGQAFOkacvI1Kf52lu4i8_cuMNYztfa--t2eGiIk5ku1bP7T7n378Q2rNZRCMz3JCXeT0EzYNMkJzlpYSiP2FiQ8">weblink (23 pages)BOOK, Applications of Computers, Session 14. Data Processing, Stuart H., Hollingdale, 1958-09-19, Atlas – Application of Computers, University of Nottingham 15–19 September 1958, Conference paper,www.chilton-computing.org.uk/acl/literature/manuals/nottingham/p014.htm,www.chilton-computing.org.uk/acl/literature/manuals/nottingham/contents.htm, 2020-05-25, live,www.chilton-computing.org.uk/acl/literature/manuals/nottingham/p014.htm," title="web.archive.org/web/20200525151358www.chilton-computing.org.uk/acl/literature/manuals/nottingham/p014.htm,">web.archive.org/web/20200525151358www.chilton-computing.org.uk/acl/literature/manuals/nottingham/p014.htm, 2020-05-25, WEB, Matthias R., Paul, Unterbrechungsfreier Schleifencode, de, Continuous loop code, 1.02,www.uni-bonn.de/~uzs180/download/mpbcd102.zip, 1995-08-10, 1994, 2008-02-11, {{cbignore}} (NB. The author called this code (English: “loop code“). It differs from Gray BCD code only in the encoding of state 0 to make it a cyclic unit-distance code for full-circle rotatory applications. Avoiding the all-zero code pattern allows for loop self-testing and to use the data lines for uninterrupted power distribution.)BOOK, Digitale Rechenautomaten – Eine Einführung, de, Digital Computers – An Introduction, Rainer, Klar, Walter de Gruyter & Co. / {{ill, G. J. Göschen’sche Verlagsbuchhandlung, de, G. J. Göschen’sche Verlagsbuchhandlung, |publication-place=Berlin, Germany |series=Sammlung Göschen |volume=1241/1241a |date=1970-02-01 |isbn=3-11-083160-0 |id=. Archiv-Nr. 7990709. |page=17 |edition=1 |url=https://books.google.com/books?id=QnqVDwAAQBAJ&pg=PA17 |access-date=2020-04-13 |url-status=live |archive-url=https://web.archive.org/web/20200601100645books.google.de/books?id=QnqVDwAAQBAJ&pg=PA17&lpg=PA17&redir_esc=y |archive-date=2020-06-01}} (205 pages) (NB. A 2019 reprint of the first edition is available under {{ISBN|3-11002793-3|978-3-11002793-8}}. A reworked and expanded REF, Klar, 1989, 4th edition, plain, exists as well.)BOOK, Digitale Rechenautomaten – Eine Einführung in die Struktur von Computerhardware, de, Digital Computers – An Introduction into the structure of computer hardware, Rainer, Klar, Walter de Gruyter & Co., Berlin, Germany, Sammlung Göschen, 2050, 1989, 1988-10-01, 3-11011700-2, 28, 4th reworked, (320 pages) (NB. The author called this code (English: “unit-distance code“). By swapping two bit rows and inverting one of them, it can be transferred into the O’Brien code II, whereas by swapping and inverting two bit rows, it can be transferred into the Petherick code.)BOOK, Notes on Analog-Digital Conversion Techniques, III.F. Unit-Distance Codes / VI.E.2. Reflected Binary Codes, Alfred Kriss, Susskind, Alfred Kriss, Susskind, John Erwin, Ward, 1958-03-28, 1957, 1956, 3, Technology Press of the Massachusetts Institute of Technology / John Wiley & Sons, Inc. / Chapman & Hall, Ltd., New York, USA, Cambridge, Massachusetts, USA, Technology Books in Science and Engineering, 1, 3-10–3-16 [3-13–3-16], 6-65–6-60 [6-60], (x+416+2 pages) (NB. The contents of the book was originally prepared by staff members of the Servomechanisms Laboraratory, Department of Electrical Engineering, MIT, for Special Summer Programs held in 1956 and 1957. Susskind’s “reading-type code” is actually a minor variant of the code shown here with the two most significant bit rows swapped to better illustrate symmetries. Also, by swapping two bit rows and inverting one of them, the code can be transferred into the Petherick code, whereas by swapping and inverting two bit rows, the code can be transferred into the O’Brien code II.)BOOK, Digitale Automaten – Theorie, Struktur, Technik, Programmieren, de, Wilhelm, Kämmerer, :de:Wilhelm Kämmerer, Hans, Frühauf, :de:Hans Frühauf, Wilhelm, Kämmerer, Kurz, Schröder, Helmut, Winkler, II.15. Struktur: Informationsdarstellung im Automaten, 1, May 1969, Akademie-Verlag GmbH, Berlin, Germany, Jena, Germany, 5, License no. 202-100/416/69. Order no. 4666 ES 20 K 3., Elektronisches Rechnen und Regeln, 173, (NB. A second edition 1973 exists as well. Like REF, Borucki, Dittmann, 1971, Borucki and Dittmann, plain, , but without naming it Glixon code, the author creates a 20-cyclic tetradic code from Glixon code and a Glixon code variant with inverted high-order bit.)BOOK, Digitale Meßtechnik: Eine Einführung, 2.3 Gebräuchliche Codes in der digitalen Meßtechnik, de, Lorenz, Borucki, Joachim, Dittmann, 1971, July 1970, 1966, Autumn 1965, Springer-Verlag, Berlin / Heidelberg, Germany, Krefeld / Karlsruhe, Germany, 2, 75-131547, 3-540-05058-2, {{ISBN, 978-3-642-80561-5, |doi=10.1007/978-3-642-80560-8 |pages=10–23 [12–14]}} (viii+252 pages) 1st edition (NB. Like REF, Kämmerer, 1969, Kämmerer, plain, , the authors describe a 6-bit 20-cyclic Glixon code.)BOOK, Design Methods for Digital Systems, 3.3. Unit Distance Codes, Jean P., Chinal, en, Akademie-Verlag / Springer-Verlag, Berlin, Germany, Paris, France, Alan, Preston, Arthur, Summer, January 1973, 1st English, License No. 202-100/542/73. Order No. 7617470(6047) ES 19 B 1 / 20 K 3, 978-0-387-05871-9, 10.1007/978-3-642-86187-1, 50, 60362404,books.google.com/books?id=8FnxCAAAQBAJ, 2020-06-21, (xviii+506 pages) (NB. The French 1967 original book was named “Techniques Booléennes et Calculateurs Arithmétiques”, published by {{ill|Éditions Dunod|fr}}.)BOOK, Military Handbook: Encoders – Shaft Angle To Digital, United States Department of Defense, MIL-HDBK-231A, 1991-09-30,everyspec.com/MIL-HDBK/MIL-HDBK-0200-0299/download.php?spec=MIL_HDBK_231A.1809.pdf, 2020-07-25, live,everyspec.com/MIL-HDBK/MIL-HDBK-0200-0299/download.php?spec=MIL_HDBK_231A.1809.pdf," title="web.archive.org/web/20200725051128everyspec.com/MIL-HDBK/MIL-HDBK-0200-0299/download.php?spec=MIL_HDBK_231A.1809.pdf,">web.archive.org/web/20200725051128everyspec.com/MIL-HDBK/MIL-HDBK-0200-0299/download.php?spec=MIL_HDBK_231A.1809.pdf, 2020-07-25, (NB. Supersedes MIL-HDBK-231(AS) (1970-07-01).)BOOK, A short note on useful codes for Fluidic Control Circuits, Codes particularly useful for analogue to digital conversions, E. Alexander, Powell, June 1968, CoA Memo 156, College of Aeronautics, Cranfield, The College of Aeronautics, Department of Production Engineering, Cranfield, UK, 215864694, 7, 9,dspace.lib.cranfield.ac.uk/bitstream/handle/1826/9559/COA_Memo_156_June_1968.pdf, 2020-12-15, live,web.archive.org/web/20201215124905/https://dspace.lib.cranfield.ac.uk/bitstream/handle/1826/9559/COA_Memo_156_June_1968.pdf, 2020-12-15, (18 pages) (NB. The paper names the Glixon code modified Gray code and misspells Richard W. Hamming’s name.)WEB, Decimal Representations, John J. G., Savard, 2018, 2006, quadibloc,www.quadibloc.com/comp/cp0203.htm, 2018-07-16, live,www.quadibloc.com/comp/cp0203.htm," title="web.archive.org/web/20180716101321www.quadibloc.com/comp/cp0203.htm,">web.archive.org/web/20180716101321www.quadibloc.com/comp/cp0203.htm, 2018-07-16, JOURNAL, Loopless Gray Code Enumeration and the Tower of Bucharest, Felix, Herter, Günter, Rote, Berlin, Germany, 2018-08-09, 2017-12, 2017-08-09, 2016-04-22, 2018-11-14, Theoretical Computer Science (journal), Theoretical Computer Science, 0304-3975, 748, 1604.06707, 4014870, 10.1016/j.tcs.2017.11.017, 40–54,page.mi.fu-berlin.de/rote/Papers/pdf/Loopless+Gray+code+enumeration+and+the+Tower+of+Bucharest.pdf, 2020-12-16, live,web.archive.org/web/20201216150701/https://page.mi.fu-berlin.de/rote/Papers/pdf/Loopless+Gray+code+enumeration+and+the+Tower+of+Bucharest.pdf, 2020-12-16, arxiv.org/pdf/1604.06707.pdf" title="https:/-/web.archive.org/web/20201216230211arxiv.org/pdf/1604.06707.pdf">weblink (15/18/19/24 pages)CONFERENCE, Using Gray codes as Location Identifiers, Thomas, Strang, Armin, Dammann, Matthias, Röckl, Simon, Plass, 6. GI/ITG KuVS Fachgespräch Ortsbezogene Anwendungen und Dienste, en, de, October 2009, Institute of Communications and Navigation, German Aerospace Center (DLR), Oberpfaffenhofen, Germany, 10.1.1.398.9164, 2020-12-16, live,elib.dlr.de/60489/3/paper.pdf,elib.dlr.de/60489/3/paper.pdf," title="web.archive.org/web/20150501063457elib.dlr.de/60489/3/paper.pdf,">web.archive.org/web/20150501063457elib.dlr.de/60489/3/paper.pdf, 2015-05-01, (5/8 pages) elib.dlr.de/60489/2/Strang_Thomas.pdf" title="https:/-/web.archive.org/web/20201216231728elib.dlr.de/60489/2/Strang_Thomas.pdf">weblink
- WEB, Thomas Strang, etal, October 2009, Using Gray codes as Location Identifiers, ResearchGate, Abstract,www.researchgate.net/publication/225003251,web.archive.org/web/20200903200722/https://www.researchgate.net/profile/Simon_Plass/publication/225003251_Using_Gray_codes_as_Location_Identifiers/links/02e7e53a733e49b2f6000000/Using-Gray-codes-as-Location-Identifiers.pdf, 2020-09-03, de, en,
BOOK, Genetic Algorithms in Search, Optimization, and Machine Learning, David Edward, Goldberg, David Edward Goldberg, Addison-Wesley, Reading, Massachusetts, USA, 1, 1989, 1989gaso.book.....G, BOOK, The Froehlich/Kent Encyclopedia of Telecommunications: Volume 2 - Batteries to Codes-Telecommunications, Fritz E., Froehlich, Allen, Kent, Allen Kent, Carolyn M., Hall, Baudot, Jean Maurice Emile, Andrew J., Butrica, Marcel Dekker Inc. / CRC Press, 1991-06-21, 2, 0-8247-2901-3, 90-3966, 31–34,books.google.com/books?id=l3Rn_VefKaoC&pg=PA31, 2020-12-20, 31, […] 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 5-element code devised by Baudot […], WEB, The Evolution of Character Codes, 1874–1968, Eric N., Fischer, 2000-06-20, ark:/13960/t07x23w8s,archive.org/details/enf-ascii, 2020-12-20, […] 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 single-unit pulse, Baudot’s system used a uniform six time units to transmit each character. […] his early telegraph probably used the six-unit code […] that he attributes to Edward Davy, Davy in an 1877 article. […] in 1876 Baudot redesigned his equipment to use a five-unit code. Punctuation and digits were still sometimes needed, though, so he adopted from David Edward Hughes, 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 five-unit 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. […], index-of.es/Varios-2/ASCII%20The%20Evolution%20of%20Character%20Codes.pdf" title="https:/-/web.archive.org/web/20180919020435index-of.es/Varios-2/ASCII%20The%20Evolution%20of%20Character%20Codes.pdf">weblinkweblinkWEB, Codes that Don’t Count - Some Printing Telegraph Codes as Products of their Technologies (With Particular Attention to the Teletypesetter), David M., MacMillan, lemur.com, Mineral Point, Wisconsin, USA, Revision 3, 2010-04-27, 2010-04-25, 2010-04-23,www.circuitousroot.com/artifice/telegraphy/tty/codes/index.html, 2020-12-20, live,www.circuitousroot.com/artifice/telegraphy/tty/codes/index.html," title="web.archive.org/web/20201218233813www.circuitousroot.com/artifice/telegraphy/tty/codes/index.html,">web.archive.org/web/20201218233813www.circuitousroot.com/artifice/telegraphy/tty/codes/index.html, 2020-12-18, BOOK, Convention télégraphique internationale de Saint-Pétersbourg et Règlement et tarifs y annexés, Revision de Lisbonne, 1908 / Extraits de la publication: Documents de la Conférence télégraphique internationale de Lisbonne, fr, 1909, 1908, Lisbon, Portugal, Berne, Switzerland, Bureau Internationale de L’Union Télégraphique, BOOK, Règlement télégraphique annexé à la convention internationale des télécommunications - protocol finale audit règlement - Madrid, 1932, fr, Chapter IX. Signaux de transmission, Article 35. Signaux de transmission des alphabets télegraphiques internationaux ‘nos 1 et 2, signaux d.u code Morse, de l’appareil Hughes et de l’appareil Siemens, Madrid, Spain, Berne, Switzerland, Bureau Internationale de L’Union Télégraphique, 1933, 1932, 31–40 [33],search.itu.int/history/HistoryDigitalCollectionDocLibrary/4.5.43.fr.201.pdf, 2020-12-21, live,search.itu.int/history/HistoryDigitalCollectionDocLibrary/4.5.43.fr.201.pdf," title="web.archive.org/web/20201221011133search.itu.int/history/HistoryDigitalCollectionDocLibrary/4.5.43.fr.201.pdf,">web.archive.org/web/20201221011133search.itu.int/history/HistoryDigitalCollectionDocLibrary/4.5.43.fr.201.pdf, 2020-12-21, (1+188 pages) www.itu.int/en/history/Pages/PlenipotentiaryConferences.aspx?conf=4.5" title="https:/-/web.archive.org/web/20201221012741www.itu.int/en/history/Pages/PlenipotentiaryConferences.aspx?conf=4.5">weblinkBOOK, Telegraph Regulations Annexed To The International Telecommunication Convention - Final Protocol To The Telegraph Regulations - Madrid 1932, Chapter IX. Transmission Signals. Article 35. Transmission Signals of the International Telegraph Alphabets Nos. 1 and 2, Morse Code Signals and Signals of the Hughes and Siemens Instruments., en, fr, 1933, 1932, 43-152-2 / 18693, 32–40 [34], General Post Office / His Majesty’s Stationery Office, London, UK,search.itu.int/history/HistoryDigitalCollectionDocLibrary/4.5.43.en.101.pdf, 2020-12-21, live,search.itu.int/history/HistoryDigitalCollectionDocLibrary/4.5.43.en.101.pdf," title="web.archive.org/web/20201221011748search.itu.int/history/HistoryDigitalCollectionDocLibrary/4.5.43.en.101.pdf,">web.archive.org/web/20201221011748search.itu.int/history/HistoryDigitalCollectionDocLibrary/4.5.43.en.101.pdf, 2020-12-21, (1+2*120+26 pages) www.itu.int/en/history/Pages/PlenipotentiaryConferences.aspx?conf=4.5" title="https:/-/web.archive.org/web/20201221012741www.itu.int/en/history/Pages/PlenipotentiaryConferences.aspx?conf=4.5">weblinkJOURNAL, Einige neuere Vorschläge zur mehrfachen Telegraphie: A. Absatzweise vielfache Telegraphie, de, 1877, (:de:Polytechnisches Journal, Dingler’s Polytechnisches Journal), Johann, Zeman, Ferdinand, Fischer, Augsburg, Germany, (:de:J. G. Cotta’sche Buchhandlung, J. G. Cotta’sche Buchhandlung), 226, 499–507,dingler.culture.hu-berlin.de/article/pj226/ar226125, 2020-12-21, live,dingler.culture.hu-berlin.de/article/pj226/ar226125," title="web.archive.org/web/20201221163504dingler.culture.hu-berlin.de/article/pj226/ar226125,">web.archive.org/web/20201221163504dingler.culture.hu-berlin.de/article/pj226/ar226125, 2020-12-21, 499, […] Der um die Mitte des J[ahres] 1874 patenti[e]rte, ebenfalls dem Henry Highton, Highton’schen verwandte Typendrucker des französischen Telegraphen-Verwaltungsbeamten Baudot wurde bei seiner 1875 patenti[e]rten Weiterentwicklung in einen fünffachen umgewandelt […], }}Further reading
- BOOK, Richard Kohler, Richards, Arithmetic Operations in Digital Computers, D. Van Nostrand Co., Inc., New York, USA, 5, 1955,books.google.com/books?id=BI5QAAAAMAAJ,
- BOOK, Richard Kohler, Richards, Electronic Digital Components and Circuits, D. Van Nostrand Co., Inc., 1967, 490, 500–504, 510–511,
- WEB, Black, Paul E., Gray code, 2004-02-25, NIST,xlinux.nist.gov/dads/HTML/graycode.html,
- BOOK, Press, William H., Teukolsky, Saul A., Vetterling, William T., Flannery, Brian P., 2007, Numerical Recipes: The Art of Scientific Computing, 3rd, Cambridge University Press, New York, USA, 978-0-521-88068-8, Section 22.3. Gray Codes,apps.nrbook.com/empanel/index.htmlpg=1166, 2011-08-18, 2011-08-11,apps.nrbook.com/empanel/index.htmlpg=1166," title="web.archive.org/web/20110811154417apps.nrbook.com/empanel/index.htmlpg=1166,">web.archive.org/web/20110811154417apps.nrbook.com/empanel/index.htmlpg=1166, dead,
- JOURNAL, Savage, Carla Diane, Carla Diane Savage, A Survey of Combinatorial Gray Codes, SIAM Review, Society for Industrial and Applied Mathematics (SIAM), 39, 4, 605–629, 1997, 10.1137/S0036144595295272,www4.ncsu.edu/~savage/AVAILABLE_FOR_MAILING/survey.ps, 2132693, 1997SIAMR..39..605S, 10.1.1.39.1924, 6375360,
- BOOK, Wilf, Herbert Saul, Herbert Saul Wilf, Chapters 1–3, Combinatorial algorithms: An update,archive.org/details/combinatorialalg0000wilf, registration, Society for Industrial and Applied Mathematics (SIAM), 1989, 0-89871-231-9,
- BOOK, Megan, Dewar, Brett, Stevens, Ordering Block Designs – Gray Codes, Universal Cycles and Configuration, Springer Science+Business Media, New York, USA, 1, 2012-08-29, 978-1-46144324-7, CMS Books in Mathematics, 1613-5237, 10.1007/978-1-4614-4325-4,
- WEB, Gray Code Fundamentals, Design How-To, Part 1, Clive “Max”, Maxfield, 2012-10-01, 2011-05-28, EETimes,www.eetimes.com/document.asp?doc_id=1278809, 2017-10-30, live,web.archive.org/web/20171030135842/https://www.eetimes.com/document.asp?doc_id=1278809, 2017-10-30, Part 2 Part 3
- BOOK, Hacker’s Delight, Hacker’s Delight, Chapter 13: Gray Code, Henry S., Warren, Jr., 2013, 2, Addison Wesley – Pearson Education, Inc., 978-0-321-84268-8, 311–317, (7 pages)
- JOURNAL, Computing Binary Combinatorial Gray Codes Via Exhaustive Search With SAT Solvers, Zinovik, Igor, Kroening, Daniel, Chebiryak, Yury, IEEE Transactions on Information Theory, IEEE, 54, 4, 2008-03-21, 1819–1823, 10.1109/TIT.2008.917695,ieeexplore.ieee.org/document/4475394, 20.500.11850/11304, 2854180, free, (5 pages)
- JOURNAL, Joseph A., O’Brien, Unit-Distance Binary-Decimal Code Translators, IRE Transactions on Electronic Computers, EC-6, 2, June 1957, 0367-9950, 10.1109/TEC.1957.5221585, 122–123,www.researchgate.net/publication/250929745, 2020-05-25, (2 pages)
- MAGAZINE, A decimal Gray code – Easily converted for shaft position coding, K. G., Barr, Faculty of Natural Sciences, University of the West Indies, March 1981, Wireless World, 87, 1542, 86–87,worldradiohistory.com/hd2/IDX-Site-Early-Radio/Archive-Wireless-World-IDX/80s/Wireless-World-1981-03-OCR-Page-0046.pdf, 2020-07-28, live,web.archive.org/web/20200728093447/https://worldradiohistory.com/hd2/IDX-Site-Early-Radio/Archive-Wireless-World-IDX/80s/Wireless-World-1981-03-OCR-Page-0046.pdf, 2020-07-28,
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