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The '''reflected binary code''' ('''RBC'''), also known as '''reflected binary''' ('''RB''') or '''Gray code''' after [[Frank Gray (researcher)|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 "{{mono|001}}", and "2" would be "{{mono|010}}". In Gray code, these values are represented as "{{mono|001}}" and "{{mono|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]] [[switch]]es and to facilitate [[error correction]] in digital communications such as [[digital terrestrial television]] and some [[DOCSIS|cable TV]] systems. The use of Gray code in these devices helps simplify logic operations and reduce errors in practice.<ref>{{Cite web |last=Gray |first=Joel |date=March 2020 |title=Understanding Gray Code: A Reliable Encoding System |url=https://graycode.ie/blog/graycode/ |access-date=2023-06-30 |website=graycode.ie |at=Section: Conclusion |language=en}}</ref>

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

: {| class="wikitable" style="text-align:center;"

|-

! Decimal !! Binary

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[[File:Reflected binary Gray 2632058.png|thumb|Gray's patent introduces the term "reflected binary code"]]

In principle, there can be more than one such code for a given word length, but the term Gray code was first applied to a particular [[binary numeral system|binary]] code for non-negative integers, the ''binary-reflected Gray code'', or '''BRGC'''. [[Bell Labs]] researcher

[[George R. &nbsp;Stibitz]] described such a code in a 1941 patent application, granted in 1943.<ref name="Stibitz_1941"/><ref name="Winder_1959"/><ref name="Knuth_2014"/> [[Frank Gray (researcher)|Frank Gray]] introduced the term ''reflected binary code'' in his 1947 patent application, remarking that the code had "as yet no recognized name".<ref name="Gray_1947"/> 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 {{nowrap|( … {{mono|11001100}} … );}} the next digit a pattern of 4 on, 4 off; the ''i''-th least significant bit a pattern of 2^''i'' on 2^''i'' off. The most significant digit is an exception to this: for an ''n''-bit Gray code, the most significant digit follows the pattern 2^''n''-1 on, 2^''n''-1 off, which is the same (cyclic) sequence of values as for the second-most significant digit, but shifted forwards 2^''n''-2 places. The four-bit version of this is shown below:

[[File:Gray_code_tesseract.svg|thumb|Visualized as a traversal of [[Vertex (graph theory)|vertices]] of a [[tesseract]]]]

[[File:Gray code number line arcs.svg|thumb|Gray code along the number line]]

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

: {| class="wikitable sortable" style="text-align:center;"

|-

! Decimal !! Binary !! Gray

|-

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It can serve as a solution guide for the [[Towers of Hanoi]] problem, based on a game by the French [[Édouard Lucas]] in 1883.<ref name="Lucas_1883"/><ref name="Parville_1883"/><ref name="Allardice-Fraser_1883"/><ref name="Lucas_1892"/> Similarly, the so-called Towers of Bucharest and Towers of Klagenfurt game configurations yield [[#n-ary|ternary and pentary]] Gray codes.<ref name="Herter-Rote_2016"/>

[[Martin Gardner]] wrote a popular account of the Gray code in his August 1972 [[List of Martin Gardner Mathematical Games columns|"Mathematical Games" column]] in ''[[Scientific American]]''.<ref name="Gardner_1972"/>

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

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=== Telegraphy codes ===

{{anchor|Baudot|Excess-1 Gray}}When the French engineer [[Émile Baudot]] changed from using a 6-unit (6-bit) code to 5-unit code for his [[printing telegraph]] system, in 1875<ref name="Zeman-Fischer_1877"/> or 1876,<ref name="Froehlich-Kent_1991"/><ref name="Fischer_2000"/> 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,<ref name="Rothen_1884"/><ref name="Pendry_1920"/><ref name="MacMillan_2010"/> and other symbols appropriately placed, the 5-bit character code has been recognized as a reflected binary code.<ref name="Knuth_2014"/> This code became known as [[Baudot code]]<ref name="ITU_1909"/> and, with minor changes, was eventually adopted as [[International Telegraph Alphabet No.&nbsp;1]] (ITA1, CCITT-1) in 1932.<ref name="ITU_1933_FR"/><ref name="ITU_1933_EN"/><ref name="MacMillan_2010"/>

{{anchor|Schäffler telegraph|Schäffler code}}About the same time, the German-Austrian {{ill|Otto Schäffler|de|Theodor Heinrich Otto Schäffler}}<ref name="Zemanek_1979"/> demonstrated another printing telegraph in Vienna using a 5-bit reflected binary code for the same purpose, in 1874.<ref name="Zemanek_1976"/><ref name="Knuth_2014"/>

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[[Frank Gray (researcher)|Frank Gray]], who became famous for inventing the signaling method that came to be used for compatible color television, invented a method to convert analog signals to reflected binary code groups using [[vacuum tube]]-based apparatus. Filed in 1947, the method and apparatus were granted a patent in 1953,<ref name="Gray_1947"/> and the name of Gray stuck to the codes. The "[[Pulse-code modulation#History|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.<ref name="Goodall_1951"/>

[[File:US02632058 Gray.png|thumb|600pxupright=1.2|center|Part of front page of Gray's patent, showing PCM tube (10) with reflected binary code in plate (15)]]

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

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=== Error correction ===

In modern [[digital communications]], 1D- and 2D-Gray codes play an important role in error prevention before applying an [[error correction]]. For example, in a [[digital modulation]] scheme such as [[quadrature amplitude modulation|QAM]] where data is typically transmitted in [[symbol rate|symbols]] of 4 bits or more, the signal's [[constellation diagram]] is arranged so that the bit patterns conveyed by adjacent constellation points differ by only '''one''' bit. By combining this with [[forward error correction]] capable of correcting single-bit errors, it is possible for a [[Receiver (radio)|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]].

<gallery heights="120" mode="packed" style="text-align:left">

QPSK Gray Coded.svg|Codes 4-PSK

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[[George R. Stibitz]] utilized a reflected binary code in a binary pulse counting device in 1941 already.<ref name="Stibitz_1941"/><ref name="Winder_1959"/><ref name="Knuth_2014"/>

A typical use of Gray code counters is building a [[FIFO (computing and electronics)|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.<ref name="Donohue_2003"/> The updated read and write pointers need to be passed between clock domains when they change, to be able to track FIFO empty and full status in each ___domain. Each bit of the pointers is sampled non-deterministically for this clock ___domain transfer. So for each bit, either the old value or the new value is propagated. Therefore, if more than one bit in the multi-bit pointer is changing at the sampling point, a "wrong" binary value (neither new nor old) can be propagated. By guaranteeing only one bit can be changing, Gray codes guarantee that the only possible sampled values are the new or old multi-bit value. Typically Gray codes of power-of-two length are used.

The updated read and write pointers need to be passed between clock domains when they change, to be able to track FIFO empty and full status in each ___domain. Each bit of the pointers is sampled non-deterministically for this clock ___domain transfer. So for each bit, either the old value or the new value is propagated. Therefore, if more than one bit in the multi-bit pointer is changing at the sampling point, a "wrong" binary value (neither new nor old) can be propagated. By guaranteeing only one bit can be changing, Gray codes guarantee that the only possible sampled values are the new or old multi-bit value. Typically Gray codes of power-of-two length are used.

Sometimes digital buses in electronic systems are used to convey quantities that can only increase or decrease by one at a time, for example the output of an event counter which is being passed between clock domains or to a digital-to-analog converter. The advantage of Gray codes in these applications is that differences in the propagation delays of the many wires that represent the bits of the code cannot cause the received value to go through states that are out of the Gray code sequence. This is similar to the advantage of Gray codes in the construction of mechanical encoders, however the source of the Gray code is an electronic counter in this case. The counter itself must count in Gray code, or if the counter runs in binary then the output value from the counter must be reclocked after it has been converted to Gray code, because when a value is converted from binary to Gray code,<ref group="nb" name="NB_Arithmetic_conversion"/> it is possible that differences in the arrival times of the binary data bits into the binary-to-Gray conversion circuit will mean that the code could go briefly through states that are wildly out of sequence. Adding a clocked register after the circuit that converts the count value to Gray code may introduce a clock cycle of latency, so counting directly in Gray code may be advantageous.<ref name="Hulst_1957"/>

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To produce the next count value in a Gray-code counter, it is necessary to have some combinational logic that will increment the current count value that is stored. One way to increment a Gray code number is to convert it into ordinary binary code,<ref name="Powell_1968"/> add one to it with a standard binary adder, and then convert the result back to Gray code.<ref name="Mehta-Owens-Irwin_1996"/> 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.<ref name="Doran_2007"/>

==== Gray code addressing <span class="anchor" id="Shifted Gray encoding"></span> ====

As the execution of [[program code]] typically causes an instruction memory access pattern of locally consecutive addresses, [[bus encoding]]s using Gray code addressing instead of binary addressing can reduce the number of state changes of the address bits significantly, thereby reducing the [[CPU power consumption]] in some low-power designs.<ref name="Su-Tsui-Despain_1994"/><ref name="Guo-Parameswaran_2010"/>

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The binary-reflected Gray code list for ''n'' bits can be generated [[recursion|recursively]] from the list for ''n''&nbsp;&minus;&nbsp;1 bits by reflecting the list (i.e. listing the entries in reverse order), prefixing the entries in the original list with a binary {{mono|0}}, prefixing the entries in the reflected list with a binary&nbsp;{{mono|1}}, and then concatenating the original list with the reversed list.<ref name="Knuth_2014"/> For example, generating the ''n''&nbsp;=&nbsp;3 list from the ''n''&nbsp;=&nbsp;2 list:

: {| cellpadding="5" border="0" style="margin: 1em;"

|-

| 2-bit list:

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

:{{block indent|{{{mono|000,001,010,011,100,101,110,111}}} → {{{mono|001,000,011,010,101,100,111,110}}} {{pad|2em}}(invert bit 0)}}

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

:{{block indent| {{{mono|000,001,010,011,100,101,110,111}}} → {{{mono|010,011,000,001,110,111,100,101}}} {{pad|2em}}(invert bit 1)}}

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

:{{block indent| {{{mono|000,001,010,011,100,101,110,111}}} → {{{mono|100,101,110,111,000,001,010,011}}} {{pad|2em}}(invert bit 2)}}

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

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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 <math>g_i</math> is the <math>i</math>th Gray-coded bit (<math>g_0</math> being the most significant bit), and <math>b_i</math> is the <math>i</math>th binary-coded bit (<math>b_0</math> being the most-significant bit), the reverse translation can be given recursively: <math>b_0 = g_0</math>, and <math>b_i=g_i \oplus b_{i-1}</math>. Alternatively, decoding a Gray code into a binary number can be described as a [[prefix sum]] of the bits in the Gray code, where each individual summation operation in the prefix sum is performed modulo two.

To construct the binary-reflected Gray code iteratively, at step 0 start with the <math>\mathrm{code}_0 = \mathtt{0}</math>, and at step <math>i > 0</math> find the bit position of the least significant {{mono|1}} in the binary representation of <math>i</math> and flip the bit at that position in the previous code <math>\mathrm{code}_{i-1}</math> to get the next code <math>\mathrm{code}_i</math>. The bit positions start 0, 1, 0, 2, 0, 1, 0, 3, ....<ref group="nb" name="NB_OEIS_A007814"/> See [[find first set]] for efficient algorithms to compute these values.

== Converting to and from Gray code ==

The following functions in [[C (programming language)|C]] convert between binary numbers and their associated Gray codes. While it may seem that Gray-to-binary conversion requires each bit to be handled one at a time, faster algorithms exist.<ref name="Dietz_2002" /><ref name="Powell_1968" /><ref group="nb" name="NB_Arithmetic_conversion" />

The first use of the function converting binary code to Gray code and writing in Gray code to binary code was developed by a student of the WSP Rzeszów in 1995. The first publication of such a function was in 1996 in Poland.

The following functions in [[C (programming language)|C]] convert between binary numbers and their associated Gray codes. While it may seem that Gray-to-binary conversion requires each bit to be handled one at a time, faster algorithms exist.<ref name="Dietz_2002" /><ref name="Powell_1968" /><ref group="nb" name="NB_Arithmetic_conversion" />

<syntaxhighlight lang="C">

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== Special types of Gray codes ==

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

However, mathematicians have discovered other kinds of Gray codes.

Like BRGCs, each consists of a list of words, where each word differs from the next in only one digit (each word has a [[Hamming distance]] of 1 from the next word).

=== Gray codes with ''n'' bits and of length less than 2^''n'' ===

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Although the binary reflected Gray code is useful in many scenarios, it is not optimal in certain cases because of a lack of "uniformity".<ref name="Bhat-Savage_1996"/> In '''balanced Gray codes''', the number of changes in different coordinate positions are as close as possible. To make this more precise, let ''G'' be an ''R''-ary complete Gray cycle having transition sequence <math>(\delta_k)</math>; the ''transition counts'' (''spectrum'') of ''G'' are the collection of integers defined by

: <math>\lambda_k = |\{ j \in \mathbb{Z}_{R^n} : \delta_j = k \}| \, , \text { for } k \in \mathbb{Z}_n</math>

: <math display="block">\lambda_k = |\{ j \in \mathbb{Z}_{R^n} : \delta_j = k \}| \, , \text { for } k \in \mathbb{Z}_n</math>

A Gray code is ''uniform'' or ''uniformly balanced'' if its transition counts are all equal, in which case we have <math>\lambda_k = \tfrac{R^n}{n}</math>

A Gray code is ''uniform'' or ''uniformly balanced'' if its transition counts are all equal, in which case we have <math>\lambda_k = \tfrac{R^n}{n}</math> for all ''k''. Clearly, when <math>R = 2</math>, such codes exist only if ''n'' is a power of 2.<ref>{{cite journal |authorfirst1=D. G. |last1=Wagner, |first2=J. |last2=West |title=Construction of Uniform Gray Codes |journal=Congressus Numerantium |volume=80 |year=1991 |pages=217–223}}</ref> If ''n'' is not a power of 2, it is possible to construct ''well-balanced'' binary codes where the difference between two transition counts is at most 2; so that (combining both cases) every transition count is either <math>2\left\lfloor \tfrac{2^n}{2n} \right\rfloor</math> or <math>2\left\lceil \tfrac{2^n}{2n} \right\rceil</math>.<ref name="Bhat-Savage_1996"/> 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.<ref name="Suparta_2005"/>

For example, a balanced 4-bit Gray code has 16 transitions, which can be evenly distributed among all four positions (four transitions per position), making it uniformly balanced:<ref name="Bhat-Savage_1996"/>

:{{block indent|1= {{mono|0 {{fontcolor|red|1}} 1 1 1 1 1 {{fontcolor|red|0}} 0 0 0 0 0 {{fontcolor|red|1}} 1 {{fontcolor|red|0}}}} }}

:{{block indent|1= {{mono|0 0 {{fontcolor|red|1}} 1 1 1 {{fontcolor|red|0}} 0 {{fontcolor|red|1}} 1 1 1 {{fontcolor|red|0}} 0 0 0}} }}

:{{block indent|1= {{mono|0 0 0 0 {{fontcolor|red|1}} 1 1 1 1 {{fontcolor|red|0}} 0 {{fontcolor|red|1}} 1 1 {{fontcolor|red|0}} 0}} }}

:{{block indent|1= {{mono|{{fontcolor|red|0}} 0 0 {{fontcolor|red|1}} 1 {{fontcolor|red|0}} 0 0 0 0 {{fontcolor|red|1}} 1 1 1 1 1}}}}

whereas a balanced 5-bit Gray code has a total of 32 transitions, which cannot be evenly distributed among the positions. In this example, four positions have six transitions each, and one has eight:<ref name="Bhat-Savage_1996"/>

:{{block indent|1= {{mono|{{fontcolor|red|1}} 1 1 1 1 {{fontcolor|red|0}} 0 0 0 {{fontcolor|red|1}} 1 1 1 1 1 {{fontcolor|red|0}} 0 {{fontcolor|red|1}} 1 1 1 1 {{fontcolor|red|0}} 0 0 0 0 0 0 0 0 0}} }}

:{{block indent|1= {{mono|0 0 0 {{fontcolor|red|1}} 1 1 1 1 1 1 1 {{fontcolor|red|0}} 0 0 0 0 0 0 {{fontcolor|red|1}} 1 1 1 1 1 {{fontcolor|red|0}} 0 0 {{fontcolor|red|1}} 1 {{fontcolor|red|0}} 0 0}}}}

:{{block indent|1= {{mono|1 1 {{fontcolor|red|0}} 0 {{fontcolor|red|1}} 1 1 {{fontcolor|red|0}} 0 0 0 0 0 {{fontcolor|red|1}} 1 1 {{fontcolor|red|0}} 0 0 {{fontcolor|red|1}} 1 1 1 1 1 {{fontcolor|red|0}} 0 0 0 0 {{fontcolor|red|1}} 1}} }}

:{{block indent|1= {{mono|1 {{fontcolor|red|0}} 0 0 0 0 0 0 {{fontcolor|red|1}} 1 1 1 1 1 {{fontcolor|red|0}} 0 0 0 0 0 {{fontcolor|red|1}} 1 1 1 1 1 1 1 {{fontcolor|red|0}} 0 0 {{fontcolor|red|1}}}}}}

:{{block indent|1= {{mono|1 1 1 1 1 1 {{fontcolor|red|0}} 0 0 0 {{fontcolor|red|1}} 1 {{fontcolor|red|0}} 0 0 0 0 0 0 0 0 {{fontcolor|red|1}} 1 {{fontcolor|red|0}} 0 0 {{fontcolor|red|1}} 1 1 1 1 1}}}}

We will now show a construction<ref name="Flahive-Bose_2007"/> and implementation<ref name="Strackx-Piessens_2016"/> for well-balanced binary Gray codes which allows us to generate an ''n''-digit balanced Gray code for every ''n''. The main principle is to inductively construct an (''n''&nbsp;+&nbsp;2)-digit Gray code <math>G'</math> given an ''n''-digit Gray code ''G'' in such a way that the balanced property is preserved. To do this, we consider partitions of <math>G = g_0, \ldots, g_{2^n-1}</math> into an even number ''L'' of non-empty blocks of the form

: <math display="block">\left\{g_0\right\}, \left\{g_1, \ldots, g_{k_2}\right\}, \left\{g_{k_2+1}, \ldots, g_{k_3}\right\}, \ldots, \left\{g_{k_{L-2}+1}, \ldots, g_{-2}\right\}, \left\{g_{-1}\right\}</math>

where <math>k_1 = 0</math>, <math>k_{L-1} = -2</math>, and <math>k_{L} \equiv -1 \pmod{2^n}</math>). This partition induces an <math>(n+2)</math>-digit Gray code given by

: <math>\mathtt{00}g_0,</math>

{{block indent|1=<math display="block">\begin{align}

: <math>\mathtt{00}g_1, \ldots, \mathtt{00}g_{k_2}, \mathtt{01}g_{k_2}, \ldots, \mathtt{01}g_1, \mathtt{11}g_1, \ldots, \mathtt{11}g_{k_2}, </math>

: <math>&\mathtt{00}g_0,</math>\\

: <math>\mathtt{11}g_{k_2+1}, \ldots, \mathtt{11}g_{k_3}, \mathtt{01}g_{k_3}, \ldots, \mathtt{01}g_{k_2+1}, \mathtt{00}g_{k_2+1}, \ldots, \mathtt{00}g_{k_3}, \ldots,</math>

: <math>&\mathtt{00}g_{-2}g_1, \mathtt{00}g_{-1}ldots, \mathtt{1000}g_{-1k_2}, \mathtt{1001}g_{-2k_2}, \ldots, \mathtt{1001}g_0g_1, \mathtt{11}g_0g_1, \mathtt{11}g_{-1}ldots, \mathtt{0111}g_{-1k_2}, \mathtt{01}g_0</math>\

: <math>&\mathtt{0011}g_{k_2+1}g_1, \ldots, \mathtt{0011}g_{k_2k_3}, \mathtt{01}g_{k_2k_3}, \ldots, \mathtt{01}g_1g_{k_2+1}, \mathtt{1100}g_{k_2+1}g_1, \ldots, \mathtt{1100}g_{k_2k_3}, </math>\ldots,\\

: <math>&\mathtt{1100}g_{k_2+1-2}, \ldotsmathtt{00}g_{-1}, \mathtt{1110}g_{k_3-1}, \mathtt{0110}g_{k_3-2}, \ldots, \mathtt{0110}g_g_0, \mathtt{k_2+111}g_0, \mathtt{0011}g_{k_2+-1}, \ldots, \mathtt{0001}g_{k_3-1}, \ldots,</math>mathtt{01}g_0

\end{align}</math>}}

If we define the ''transition multiplicities''

: <math display="block">m_i = \left|\left\{ j : \delta_{k_j} = i, 1 \leq j \leq L \right\}\right|</math>

to be the number of times the digit in position ''i'' changes between consecutive blocks in a partition, then for the (''n''&nbsp;+&nbsp;2)-digit Gray code induced by this partition the transition spectrum <math>\lambda'_i</math> is

: <math>

<math display="block">

\lambda'_i = \begin{cases}

4 \lambda_i - 2 m_i, & \text{if } 0 \leq i < n \\

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

: <math display="block">V_n(i) = \{ v \in V_n : v \text{ has weight } i \}</math>

for <math>0 \leq i \leq n</math>. These levels satisfy <math>|V_n(i)| = \textstyle\binom{n}{i}</math>. Let <math>Q_n(i)</math> be the subgraph of <math>Q_n</math> induced by <math>V_n(i) \cup V_n(i+1)</math>, and let <math>E_n(i)</math> be the edges in <math>Q_n(i)</math>. A monotonic Gray code is then a Hamiltonian path in <math>Q_n</math> such that whenever <math>\delta_1 \in E_n(i)</math> comes before <math>\delta_2 \in E_n(j)</math> in the path, then <math>i \leq j</math>.

An elegant construction of monotonic ''n''-digit Gray codes for any ''n'' is based on the idea of recursively building subpaths <math>P_{n,j}</math> of length <math>2 \textstyle\binom{n}{j}</math> having edges in <math>E_n(j)</math>.<ref name="Savage-Winkler_1995"/> We define <math>P_{1,0} = (\mathtt{0}, \mathtt{1})</math>, <math>P_{n,j} = \emptyset</math> whenever <math>j < 0</math> or <math>j \geq n</math>, and

: <math>

<math display="block">

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

</math>

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

: <math>

<math display="block">

G_n^{(1)} = P_{n,0} P_{n,1}^R P_{n,2} P_{n,3}^R \cdots \text{ and } G_n^{(2)} = P_{n,0}^R P_{n,1} P_{n,2}^R P_{n,3} \cdots

</math>

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

: {| class="wikitable floatright" style="text-align: center;"

|+ Subpaths in the Savage–Winkler algorithm

|-

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An STGC for ''P''&nbsp;=&nbsp;30 and ''n''&nbsp;=&nbsp;5 is reproduced here:

: {|class="wikitable" style="text-align:center; background:#FFFFFF; border-width:0;"

|+ Single-track Gray code for 30 positions

! Angle || Code

Line 488 ⟶ 498:

|-

| 60° || {{mono|11000}} || 132° || {{mono|01100}} || 204° || {{mono|00110}} || 276° || {{mono|00011}} || 348° || {{mono|10001}}

|}<!--

<!--

But even better would be an actual illustration of the single track, on a rotary encoder, with the 5 pick-up sensors.

This is done – does that mean that the 1s and 0s can go now? They take up a whole lot of visual space.

Line 500 ⟶ 509:

[[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,<ref name="Experts_Exchange_Williams_2008">{{cite web |last1=Williams |first1=Gary |title='single track gray code' sought for encoding 360 degrees with 9 sensors |url=https://www.experts-exchange.com/questions/23594359/'single-track-gray-code'-sought-for-encoding-360-degrees-with-9-sensors.html#a22127642 |website=Experts Exchange|date=25 July 2008 }}</ref>{{ugc|date=January 2025}} based on previous work,<ref name="Etzion-Paterson_1996"/> discovered a 9-bit single track Gray code that gives a 1 degree resolution. This Gray code was used to design an actual device which was published on the site [[Thingiverse]]. This device<ref name="Thingiverse_9-Bit_Singletrack_Gray_Code_Rotorary_Encoder">{{cite web |last1=Bauer |first1=Florian |title=9-Bit Absolute Singletrack Gray Code Rotary Encoder |url=https://www.thingiverse.com/thing:4590077 |website=Thingiverse}}</ref> was designed by etzenseep (Florian Bauer) in September 2022.

<ref name="Etzion-Paterson_1996"></ref> discovered a 9-bit Single Track Gray Code that gives a 1 degree resolution. This gray code was used to design an actual device which was published on the site [[Thingiverse]]. This device<ref name="Thingiverse_9-Bit_Singletrack_Gray_Code_Rotorary_Encoder">{{cite web |last1=Bauer |first1=Florian |title=9-Bit Absolute Singletrack Gray Code Rotary Encoder |url=https://www.thingiverse.com/thing:4590077 |website=Thingiverse}}</ref> was designed by etzenseep (Florian Bauer) in September, 2022.

An STGC for ''P''&nbsp;=&nbsp;360 and ''n''&nbsp;=&nbsp;9 is reproduced here:

: {|class="wikitable" style="text-align:center; background:#FFFFFF; border-width:0;"

|+ Single-track Gray code for 360 positions

! Angle || Code

Line 925 ⟶ 933:

| 359° || {{mono||100000000}}

|}

: {|class="wikitable" style="text-align:center; background:#FFFFFF; border-width:0;"

|+ Starting and ending angles for the 20 tracks for a Singlesingle-track Gray Codecode with 9 sensors separated by 40°

! Starting Angleangle || Ending Angleangle || Length

|rowspan="21" style="text-align:center; background:#FFFFFF; border-width:0;"|

|-

Line 979 ⟶ 987:

Two-dimensional Gray codes also have uses in [[___location identification]]s 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.<ref name="Strang-Dammann-Roeckl-Plass_2009"/>

=== Excess- Gray- code ===

If a subsection of a specific codevalue is extracted from that value, for example the last 3 bits of a 4-bit gray-Gray code, the resulting code will be an "excess grayGray 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 grayGray-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 grayGray code, 7, is encoded as (0)100. Adding 1 results in number 8, encoded in grayGray 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, grayGray-encoded values in a serial fashion, one should therefore pay attention whether the sensor produces those multiple values encoded in 1 single gray-Gray code or as separate ones, as otherwise the values might appear to be counting backwards when an "overflow" is expected.

== Gray isometry ==

Line 996 ⟶ 1,004:

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

* {{anchor|Datex}}Datex codes akaor Giannini codes (1954), as described by Carl P. Spaulding,<ref name="Spaulding_1954"/><ref name="Spaulding_1965"/><ref name="Wheeler_1969"/><ref name="Dokter-Steinhauer_1973"/><ref name="Dokter-Steinhauer_1975"/><ref name="MIL_1991"/> use a variant of [[#O'Brien II|O'Brien code&nbsp;II]].

* {{anchor|Varec}} Codes used by Varec (cac. 1954),<ref name="Varec_1954"/><ref name="Bishup-Repeta-Giarrizzo_1963"/><ref name="Whessoe_1993"/><ref name="Emerson"/> use a variant of [[#O'Brien I|O'Brien code I]] as well as base-12 and base-16 Gray code variants.

* {{anchor|Lucal|MRB}}Lucal code (1959)<ref name="Lucal_1959"/><ref name="Sellers-Hsiao-Bearnson_1968"/><ref name="Doran_2007"/> aka modified reflected binary code (MRB)<ref name="Lucal_1959"/><ref name="Sellers-Hsiao-Bearnson_1968"/><ref group="nb" name="NB_Modified"/>

* {{anchor|Gillham}}[[Gillham code]] (1961/1962),<ref name="Wheeler_1969"/><ref name="Wightman_1972"/><ref name="MIL_1991"/><ref name="Phillips_1998"/><ref name="Stewart_2010"/> uses a variant of [[#Datex|Datex]] code and [[#O'Brien II|O'Brien code&nbsp;II]].

Line 1,018 ⟶ 1,026:

| colspan="18"|

|-

| rowspan="4"| {{anchor|Gray BCD}}'''Gray&nbsp;BCD''' || style="background:lightgray"|'''4''' || {{mono|0}} || {{mono|0}} || {{mono|0}} || {{mono|0}} || {{mono|0}} || {{mono|0}} || {{mono|0}} || {{mono|0}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || rowspan="4"|0—30–3 || rowspan="4"|4 (3<ref group="nb" name="NB_Tracks_Inverse"/>) || rowspan="4"|No || rowspan="4"|(2, 4, 8, 16) || rowspan="4"|No || rowspan="4"|<ref name="Steinbuch_1962"/><ref name="Steinbuch-Weber-Heinemann_1974"/>

|-

| style="background:lightgray"|'''3''' || {{mono|0}} || {{mono|0}} || {{mono|0}} || {{mono|0}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}}

Line 1,028 ⟶ 1,036:

| colspan="18"|

|-

| rowspan="4"| {{anchor|Paul}}'''Paul''' || style="background:lightgray"|'''4''' || style="background:#0FF"|{{mono|1}} || {{mono|0}} || {{mono|0}} || {{mono|0}} || {{mono|0}} || {{mono|0}} || {{mono|0}} || {{mono|0}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || rowspan="4"|1—31–3 || rowspan="4"|4 (3<ref group="nb" name="NB_Tracks_Inverse"/>) || rowspan="4"|No || rowspan="4"|2, 10 || rowspan="4"|No || rowspan="4"|<ref name="Paul_1995"/>

|-

| style="background:lightgray"|'''3''' || {{mono|0}} || {{mono|0}} || {{mono|0}} || {{mono|0}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}}

Line 1,038 ⟶ 1,046:

| colspan="18"|

|-

| rowspan="4"|'''Glixon''' || style="background:lightgray"|'''4''' || {{mono|0}} || {{mono|0}} || {{mono|0}} || {{mono|0}} || {{mono|0}} || {{mono|0}} || {{mono|0}} || {{mono|0}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || rowspan="4"|0—30–3 || rowspan="4"|4 || rowspan="4"|No || rowspan="4"|2, 4, 8, 10 || rowspan="4"|(shifted +1) || rowspan="4"|<ref name="Glixon_1957"/><ref name="Steinbuch_1962"/><ref name="Steinbuch-Weber-Heinemann_1974"/><ref name="Borucki-Dittmann_1971"/><ref name="Kämmerer_1969"/><ref group="nb" name="NB_OBrien-I_Glixon"/>

|-

| style="background:lightgray"|'''3''' || {{mono|0}} || {{mono|0}} || {{mono|0}} || {{mono|0}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || {{mono|0}}

Line 1,048 ⟶ 1,056:

| colspan="18"|

|-

| rowspan="4"|'''Tompkins&nbsp;I''' || style="background:lightgray"|'''4''' || {{mono|0}} || {{mono|0}} || {{mono|0}} || {{mono|0}} || {{mono|0}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || rowspan="4"|0—40–4 || rowspan="4"|2 || rowspan="4"|No || rowspan="4"|2, 4, 10 || rowspan="4"|Yes || rowspan="4"|<ref name="Tompkins_1956"/><ref name="Steinbuch_1962"/><ref name="Steinbuch-Weber-Heinemann_1974"/>

|-

| style="background:lightgray"|'''3''' || {{mono|0}} || {{mono|0}} || {{mono|0}} || {{mono|0}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || {{mono|0}}

Line 1,058 ⟶ 1,066:

| colspan="18"|

|-

| rowspan="4"|'''O'Brien&nbsp;I''' '''(Watts)''' || style="background:lightgray"|'''4''' || {{mono|0}} || {{mono|0}} || {{mono|0}} || {{mono|0}} || {{mono|0}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || rowspan="4"|0—30–3 || rowspan="4"|4 || rowspan="4"|9<ref name="Hoklas_2005_1"/><ref name="Hoklas_2005_2"/><ref group="nb" name="NB_Complement_InvertTop"/> || rowspan="4"|2, 4, 10 || rowspan="4"|Yes || rowspan="4"|<ref name="O'Brien_1955"/><ref name="Steinbuch_1962"/><ref name="Steinbuch-Weber-Heinemann_1974"/><ref group="nb" name="NB_OBrien-I_Glixon"/>

|-

| style="background:lightgray"|'''3''' || {{mono|0}} || {{mono|0}} || {{mono|0}} || {{mono|0}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || {{mono|0}} || {{mono|0}} || {{mono|0}} || {{mono|0}}

Line 1,068 ⟶ 1,076:

| colspan="18"|

|-

| rowspan="4"|'''Petherick''' '''(RAE)''' || style="background:lightgray"|'''4''' || {{mono|0}} || {{mono|0}} || {{mono|0}} || {{mono|0}} || {{mono|0}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || rowspan="4"|1—31–3 || rowspan="4"|3 || rowspan="4"|9<ref name="Hoklas_2005_1"/><ref name="Hoklas_2005_2"/><ref group="nb" name="NB_Complement_InvertTop"/> || rowspan="4"|2, 10 || rowspan="4"|Yes || rowspan="4"|<ref name="Petherick_1953"/><ref name="Charnley-Bidgood-Boardman_1965"/><ref group="nb" name="NB_Petherick_OBrien-II"/>

|-

| style="background:lightgray"|'''3''' || style="background:#0FF"|{{mono|1}} || {{mono|0}} || {{mono|0}} || {{mono|0}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || {{mono|0}} || {{mono|0}} || {{mono|0}} || style="background:#0FF"|{{mono|1}}

Line 1,078 ⟶ 1,086:

| colspan="18"|

|-

| rowspan="4"|'''O'Brien&nbsp;II''' || style="background:lightgray"|'''4''' || {{mono|0}} || {{mono|0}} || {{mono|0}} || {{mono|0}} || {{mono|0}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || rowspan="4"|1—31–3 || rowspan="4"|3 || rowspan="4"|9<ref name="Dokter-Steinhauer_1973"/><ref name="Hoklas_2005_1"/><ref name="Hoklas_2005_2"/><ref group="nb" name="NB_Complement_InvertTop"/> || rowspan="4"|2, 10 || rowspan="4"|Yes || rowspan="4"|<ref name="O'Brien_1955"/><ref name="Steinbuch_1962"/><ref name="Steinbuch-Weber-Heinemann_1974"/><ref group="nb" name="NB_Petherick_OBrien-II"/>

|-

| style="background:lightgray"|'''3''' || {{mono|0}} || {{mono|0}} || {{mono|0}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || {{mono|0}} || {{mono|0}} || {{mono|0}}

Line 1,088 ⟶ 1,096:

| colspan="18"|

|-

| rowspan="4"|{{anchor|Susskind}}'''Susskind<!--&nbsp;I-->''' || style="background:lightgray"|'''4''' || {{mono|0}} || {{mono|0}} || {{mono|0}} || {{mono|0}} || {{mono|0}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || rowspan="4"|1—41–4 || rowspan="4"|3 || rowspan="4"|9<ref group="nb" name="NB_Complement_InvertTop"/> || rowspan="4"|2, 10 || rowspan="4"|Yes || rowspan="4"|<ref name="Susskind_1958"/>

|-

| style="background:lightgray"|'''3''' || {{mono|0}} || {{mono|0}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || {{mono|0}} || {{mono|0}}

Line 1,098 ⟶ 1,106:

| colspan="18"|

|-

| rowspan="4"|{{anchor|Klar}}'''Klar''' || style="background:lightgray"|'''4''' || {{mono|0}} || {{mono|0}} || {{mono|0}} || {{mono|0}} || {{mono|0}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || rowspan="4"|0—40–4 || rowspan="4"|4 (3<ref group="nb" name="NB_Tracks_Inverse"/>) || rowspan="4"|9<ref group="nb" name="NB_Complement_InvertTop"/> || rowspan="4"|2, 10 || rowspan="4"|Yes || rowspan="4"|<ref name="Klar_1970"/><ref name="Klar_1989"/>

|-

| style="background:lightgray"|'''3''' || {{mono|0}} || {{mono|0}} || {{mono|0}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || {{mono|0}} || {{mono|0}} || {{mono|0}}

Line 1,108 ⟶ 1,116:

| colspan="18"|

|-

| rowspan="4"|'''Tompkins&nbsp;II''' || style="background:lightgray"|'''4''' || {{mono|0}} || {{mono|0}} || {{mono|0}} || {{mono|0}} || {{mono|0}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || rowspan="4"|1—31–3 || rowspan="4"|2 || rowspan="4"|9<ref group="nb" name="NB_Complement_Tompkins"/> || rowspan="4"|2, 10 || rowspan="4"|Yes || rowspan="4"|<ref name="Tompkins_1956"/><ref name="Steinbuch_1962"/><ref name="Steinbuch-Weber-Heinemann_1974"/>

|-

| style="background:lightgray"|'''3''' || {{mono|0}} || {{mono|0}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || {{mono|0}} || {{mono|0}} || {{mono|0}}

Line 1,118 ⟶ 1,126:

| colspan="18"|

|-

| rowspan="4"|{{nowrap|'''Excess-3 Gray'''}} || style="background:lightgray"|'''4''' || {{mono|0}} || {{mono|0}} || {{mono|0}} || {{mono|0}} || {{mono|0}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || rowspan="4"|1—41–4 || rowspan="4"|4 || rowspan="4"|9<ref name="Hoklas_2005_1"/><ref name="Hoklas_2005_2"/><ref group="nb" name="NB_Complement_InvertTop"/> || rowspan="4"|2, 10 || rowspan="4"|Yes || rowspan="4"|<ref name="MIL_1991"/><ref name="Hoklas_2005_1"/>

|-

| style="background:lightgray"|'''3''' || {{mono|0}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || style="background:#0FF"|{{mono|1}} || {{mono|0}}

Line 1,158 ⟶ 1,166:

<ref name="Lucas_1883">{{cite book |title=La tour d'Hanoï: Véritable casse tête annamite - Jeu rapporté du Tonkin par le Professeur N. Claus (de Siam) Mandarin du Collège Li Sou Stian! |language=fr |author-first=Édouard |author-last=Lucas |author-link=Édouard Lucas |date=November 1883 |publisher=Imprimerie Paul Bousrez, Tours.}} (NB. N.&nbsp;Claus de Siam is an anagram of Lucas d'Amiens, pseudonym of the author [[Édouard Lucas]].)</ref>

<ref name="Parville_1883">{{cite journal |title=La tour d'Hanoï, véritable casse-tête annamite, jeu rapporté du Tonkin par le professeur N. Claus (de Siam), mandarin du collège Li-Sou-Stian. Un vrai casse-tête, en effet, mais intéressant. Nous ne saurions mieux remercier le mandarin de son aimable intention à l'égard d'un profane qu'en signalant la Tour d'Hanoï aux personnes patientes possédées par le démon du jeu. |language=fr |editor-first=Henri |editor-last=de Parville |editor-link=:fr:Henri de Parville |date=1883-12-27 |journal=[[Journal des Débats Politiques et Littéraires]] |series=Revue des science |type=Review |edition=Matin |publication-place=Paris, France |id=ark:/12148/bpt6k462461g |pages=1–2 [2] |url=https://gallica.bnf.fr/ark:/12148/bpt6k462461g/f2.item |access-date=2020-12-18 |url-status=live |archive-url=https://web.archive.org/web/20201218125345/https://gallica.bnf.fr/ark:/12148/bpt6k462461g/f2.item |archive-date=2020-12-18}} (1 page)</ref>

<ref name="Allardice-Fraser_1883">{{cite journal |title=La Tour d'Hanoï |language=en, fr |editor-first1=Robert Edgar |editor-last1=Allardice |editor-link1=Robert Edgar Allardice |editor-first2=Alexander Yule |editor-last2=Fraser |editor-link2=Alexander Yule Fraser |journal=[[:nl:Proceedings of the Edinburgh Mathematical Society|Proceedings of the Edinburgh Mathematical Society]]<!-- :d:Q15492536 --> |volume=2 |number=5 |date=February 1883 |pages=50–53 |doi=10.1017/S0013091500037147 |issn=0013-0915 |eissn=1464-3839 |publisher=[[Edinburgh Mathematical Society]] |last1=Allardice |first1=R. E. |last2=Fraser |first2=A. Y. |doi-broken-date=1 November 2024 |s2cid=122159381 |doi-access=free }} [https://web.archive.org/web/20201218112132/https://www.cambridge.org/core/services/aop-cambridge-core/content/view/082EFE016BF3313A7BAA9335A9C0BCC1/S0013091500037147a.pdf/la-tour-d-hanoi.pdf] (4 pages)</ref>

<ref name="Lucas_1892">{{cite book |author-first=Édouard |author-last=Lucas |author-link=Édouard Lucas |title=Récréations mathématiques |language=fr |volume=3 |orig-date=1892 |edition=Librairie Albert Blanchard reissue |date=1979 |page=58}} (The first edition of this book was published post-humously.)</ref>

<ref name="Gros_1872">{{cite book |title=Théorie du baguenodier par un clerc de notaire lyonnais |language=fr |author-first=Luc-Agathon-Louis |author-last=Gros |date=1872 |publisher=[[Aimé Vingtrinier]] |edition=1 |publication-place=Lyon, France |url=https://books.google.com/books?id=EcoBJRekd-sC&pg=PP1 |access-date=2020-12-17 |url-status=live |archive-url=http://archive.wikiwix.com/cache/20170403080613/https://books.google.fr/books?id=EcoBJRekd-sC&pg=PP1 |archive-date=2017-04-03}} [https://web.archive.org/web/20201217202507/https://books.googleusercontent.com/books/content?req=AKW5Qacg5gq7SW9L1sUUqK7LQ0yJLqaCu90GFZZYzcH7dsoifr-n9hxiB30SJ1mXq3FhDejHQ7fXY2ZdzlhFywe8pQGNTgMHjX0ANYxRohedbG1FmoPTuKibswKOO7qS2X4MUwQP0dBn-vt3dfTIBFRvTBOVoZZA1ROBXTGTMybnA4gGndl8v0qIAeKsRNUuEddqabMqMngisr8fp-iNt7NMFfwq_tPk0we0yXNJWj74AFsFr1JGOlAmdQ6B_OJVTOj33IhoYybhf8LMzfpaHusXsY2BWT2ZwVQulW0mgBaem9xvRZ58TSw](2+16+4 pages and 4 pages foldout) (NB. This booklet was published anonymously, but is known to have been authored by Louis Gros.)</ref>

Line 1,184 ⟶ 1,192:

<ref name="Richards_1971">{{cite book |author-first=Richard Kohler |author-last=Richards |title=Digital Design |chapter=Snake-in-the-Box Codes |publisher=[[Wiley-Interscience]], [[John Wiley & Sons, Inc.]] |___location=Ames, Iowa, USA |publication-place=New York, USA |date=January 1971 |isbn=0-471-71945-5 |lccn=73-147235 |pages=206–207}} (12+577+1 pages)</ref>

<ref name="Sellers-Hsiao-Bearnson_1968">{{cite book |author-first1=Frederick F. |author-last1=Sellers, Jr.<!-- to be sync'ed with corresponding citeref --> |author-first2=Mu-Yue |author-last2=Hsiao |author-first3=Leroy W. |author-last3=Bearnson |title=Error Detecting Logic for Digital Computers |publisher=[[McGraw-Hill Book Company]] |___location=New York, USA |edition=1st |oclc=439460 |lccn=68-16491 |pages=152–164 |date=November 1968}}</ref>

<ref name="Lucal_1959">{{cite journal |author-first=Harold M. |author-last=Lucal |title=Arithmetic Operations for Digital Computers Using a Modified Reflected Binary |journal=[[IRE Transactions on Electronic Computers]] |volume=EC-8 |number=4 |pages=449–458 |date=December 1959 |issn=0367-9950 |doi=10.1109/TEC.1959.5222057 |s2cid=206673385 |url=https://ieeexplore.ieee.org/document/5222057}} (10 pages)</ref>

<ref name="Spaulding_1954">{{cite web |title=Digital coding and translating system |author-first=Carl P. |author-last=Spaulding |publisher=Datex Corporation |___location=Monrovia, California, USA |year=1965a<!-- necessary for proper citeref linking --> |date=1965-01-12<!-- gdate --> |orig-date=1954-03-09<!-- fdate --> |id={{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/20200805101618/https://patentimages.storage.googleapis.com/7f/1d/09/6a9b1fa3e67cb8/US3165731.pdf |archive-date=2020-08-05}} (28 pages)</ref>

<ref name="Spaulding_1965">{{cite book |title=How to Use Shaft Encoders |author-first=Carl P. |author-last=Spaulding |year=1965b<!-- necessary for proper citeref linking --> |date=1965-07-12 |publisher=Datex Corporation<!-- Datex Div, of Conrac Corp. / a subsidiary of Giannini Control Corp. --> |___location=Monrovia, California, USA}} (85 pages)</ref>

Line 1,194 ⟶ 1,202:

<ref name="Steinbuch_1962">{{cite book |title=Taschenbuch der Nachrichtenverarbeitung |language=de |editor-first=Karl W. |editor-last=Steinbuch |editor-link=Karl W. Steinbuch |date=1962 |edition=1 |publisher=[[Springer-Verlag OHG]] |___location=Karlsruhe, Germany |publication-place=Berlin / Göttingen / New York |lccn=62-14511 |pages=71–74, 97, 761–764, 770, 1080–1081}}</ref>

<ref name="Steinbuch-Weber-Heinemann_1974">{{cite book |title=Taschenbuch der Informatik – Band II – Struktur und Programmierung von EDV-Systemen |language=de |editor-first1=Karl W. |editor-last1=Steinbuch |editor-link1=Karl W. Steinbuch |editor-first2=Wolfgang |editor-last2=Weber <!-- |editor-link2=:de:Wolfgang Weber (Ingenieur)? --> |editor-first3=Traute |editor-last3=Heinemann |date=1974 |orig-date=1967 |edition=3 |volume=2 |series=Taschenbuch der Nachrichtenverarbeitung |publisher=[[Springer Verlag]] |___location=Berlin, Germany |isbn=3-540-06241-6 |lccn=73-80607 |pages=98–100}}</ref>

<ref name="Dokter-Steinhauer_1973">{{cite book |title=Digital Electronics |author-first1=Folkert |author-last1=Dokter |author-first2=Jürgen |author-last2=Steinhauer |chapter=2.4. Coding numbers in the binary system |date=1973-06-18 |series=Philips Technical Library (PTL) / Macmillan Education |publisher=[[The Macmillan Press Ltd.]] / [[N. V. Philips' Gloeilampenfabrieken]] |edition=Reprint of 1st English |___location=Eindhoven, Netherlands |sbn=333-13360-9 |isbn=978-1-349-01419-4 |doi=10.1007/978-1-349-01417-0 |pages=32, 39, 50–53 |url=https://books.google.com/books?id=hlRdDwAAQBAJ |access-date=2020-05-11 |quote-page=53 |quote=[…] 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)</ref>

<ref name="Dokter-Steinhauer_1975">{{cite book |author-first1=Folkert |author-last1=Dokter |author-first2=Jürgen |author-last2=Steinhauer |title=Digitale Elektronik in der Meßtechnik und Datenverarbeitung: Theoretische Grundlagen und Schaltungstechnik |chapter=2.4.4.6. Einschrittige Kodes |language=de |series=Philips Fachbücher |publisher=[[Deutsche Philips GmbH]] |publication-place=Hamburg, Germany |volume=I |date=1975 |orig-date=1969 |edition=improved and extended 5th |isbn=3-87145-272-6 |pages=41, 48, 51, 58, 60–61}} (xii+327+3 pages)</ref>

<ref name="Petherick_1953">{{cite book |author-first=Edward John |author-last=Petherick |title=A Cyclic Progressive Binary-coded-decimal System of Representing Numbers |date=October 1953 |type=Technical Note MS15 |publisher=[[Royal Aircraft Establishment]] (RAE) |___location=Farnborough, UK}} (4 pages) (NB. Sometimes referred to as ''A Cyclic-Coded Binary-Coded-Decimal System of Representing Numbers''.)</ref>

<ref name="Petherick-Hopkins_1958">{{cite book |author-first1=Edward John |author-last1=Petherick |author-first2=A. J. |author-last2=Hopkins |title=Some Recently Developed Digital Devices for Encoding the Rotations of Shafts |date=1958 |type=Technical Note MS21 |publisher=[[Royal Aircraft Establishment]] (RAE) |___location=Farnborough, UK}}</ref>

<ref name="Baumgartner_1963">{{cite periodical |title=Digitizer als Analog-Digital-Wandler in der Steuer-, Meß- und Regeltechnik |periodical=Technische Mitteilungen |series=Relais, elektronische Geräte, Steuerungen |language=de |date=May 1963 |issue=13 |publisher=Franz Baumgartner (FraBa) |___location=Cologne-Niehl, Germany |pages=1–2 |url=https://www.posital.com/media/posital_media/documents/TechnischeMitteilung_May1963.pdf |access-date=2020-05-21 |archive-url=https://web.archive.org/web/20200521211656/https://www.posital.com/media/posital_media/documents/TechnischeMitteilung_May1963.pdf |archive-date=2020-05-21 |quote-pages=1–2 |quote=[…] 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 [[#Petherick|Petherick-Code]]. […]}} (4 pages)</ref>

<ref name="Charnley-Bidgood-Boardman_1965">{{cite journal |title=The Design of a Pneumatic Position Encoder |author-first1=C. J. |author-last1=Charnley |author-first2=R. E. |author-last2=Bidgood |author-first3=G. E. T. |author-last3=Boardman |journal=IFAC Proceedings Volumes |publisher=The College of Aeronautics, Cranfield, Bedford, England |volume=2 |issue=3 |date=October 1965 |pages=75–88 |id=Chapter 1.5. |doi=10.1016/S1474-6670(17)68955-9 |url=https://ac.els-cdn.com/S1474667017689559/1-s2.0-S1474667017689559-main.pdf?_tid=f0c1e48e-f95b-11e7-ad9a-00000aab0f01&acdnat=1515956073_9006e89e176c6a840b5454c38525240b |access-date=2018-01-14 }}{{Dead link|date=October 2023 |bot=InternetArchiveBot |fix-attempted=yes }}</ref>

<ref name="Wheeler_1969">{{cite book |title=Analog to digital encoder |author-first=Edwin L. |author-last=Wheeler |publisher=Conrac Corporation |___location=New York, USA |date=1969-12-30<!-- gdate --> |orig-date=1968-04-05<!-- fdate --> |id={{US patent|3487460A}}. Serial No. 719026 (397812<!-- 1964-09-21 -->) |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/20200805102804/https://patentimages.storage.googleapis.com/f0/c0/60/9c3231f7e8ed44/US3487460.pdf |archive-date=2020-08-05 |quote-page=5<!-- of text, p. 8 in total -->, left column 9, rows 15–22 |quote=[…] The [[MOA-Gillham code|MOA-GILLHAM code]] is essentially the combination of the Gray code discussed thereinabove and the well known [[#Datex|Datex code]]; the Datex code is disclosed in U.S. Patent {{citeref|Spaulding|1965a|3,165,731|style=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)</ref>

<ref name="Bishup-Repeta-Giarrizzo_1963">{{cite web |title=Telemetering and supervisory control system having normally continuous telemetering signals |id=US3397386A |author-first1=Bernard W. |author-last1=Bishup |author-first2=Anthony A. |author-last2=Repeta |author-first3=Frank C. |author-last3=Giarrizzo |publisher=[[Leeds & Northrup|Leeds and Northrup Co.]] |date=1968-08-13 |orig-date=1963-04-03 |url=https://patents.google.com/patent/US3397386A/en}} [https://patentimages.storage.googleapis.com/c8/f3/cc/936fef75655a2c/US3397386.pdf]</ref>

<ref name="Savage_1997">{{cite journal |author-first=Carla Diane |author-last=Savage |author-link=Carla Diane Savage |title=Long cycles in the middle two levels of the Boolean lattice |journal=[[Ars Combinatoria (journal)|Ars Combinatoria]] |issn=0381-7032 |volume=35 |issue=A |date=1997-01-16 |___location=North Carolina State University, Raleigh, North Carolina, USA |s2cid=15975960 |citeseerx=10.1.1.39.2249 |pages=97–108 |url=http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=DACA73A3D791549CE05443B7948527EC?doi=10.1.1.39.2249&rep=rep1&type=pdf |access-date=2020-05-13 |url-status=live |archive-url=https://web.archive.org/web/20200513191647/http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=DACA73A3D791549CE05443B7948527EC?doi=10.1.1.39.2249&rep=rep1&type=pdf |archive-date=2020-05-13}} (15 pages)</ref>

<ref name="Phillips_1998">{{cite web |title=Altitude – MODEC ASCII |author-first=Darryl |author-last=Phillips |date=2012-07-26 |orig-date=1998 |publisher=AirSport Avionics |url=http://www.airsport-corp.com/modecascii.txt |url-status=usurped |archive-url=https://web.archive.org/web/20120726003224/http://www.airsport-corp.com/modecascii.txt |archive-date=2012-07-26}}</ref>

<ref name="Stewart_2010">{{cite web |title=Aviation Gray Code: Gillham Code Explained |date=2010-12-03 |author-first=K. |author-last=Stewart |publisher=Custom Computer Services (CCS) |url=httphttps://www.ccsinfo.com/forum/viewtopic.php?p=140960#140960 |access-date=2018-01-14 |url-status=live |archive-url=https://web.archive.org/web/20180116184525/http://www.ccsinfo.com/forum/viewtopic.php?p=140960 |archive-date=2018-01-16}}</ref>

<ref name="Leslie-Russell_1964">{{cite book |title=A cyclic progressive decimal code for simple translation to decimal and analogue outputs |author-first1=William "Bill" H. P. |author-last1=Leslie |author-first2=A. |author-last2=Russell |publisher=[[National Engineering Laboratory]] |___location=East Kilbride, Glasgow, UK |date=1964 |type=Report |id=NEL Report 129}} (17 pages)</ref>

<ref name="Russell_1964">{{cite journal |title=Some Binary Codes and a Novel Five-Channel Code |author-first=A. |author-last=Russell |date=August 1964 |journal=Control (Systems, Instrumentation, Data Processing, Automation, Management, incorporating Automation Progress)<!-- "Control" (London) is a predecessor of "Control and Instrumentation" (C&I), ISSN 0010-8022, formed in April 1969 by merger with "Measurement and Instrument Review" (M&I). --> |volume=8 |number=74 |series=Special Features |publisher=Morgan-Grampain (Publishers) Limited |publication-place=London, UK |pages=399–404 |url=https://books.google.com/books?id=jf4eAAAAMAAJ |access-date=2020-06-22}} (6 pages)</ref>

Line 1,224 ⟶ 1,232:

<ref name="Edwards_2004">{{cite book |title=Cogwheels of the Mind: The Story of Venn Diagrams |author-first=Anthony William Fairbank |author-last=Edwards |author-link=Anthony William Fairbank Edwards |publisher=[[Johns Hopkins University Press]] |date=2004 |isbn=0-8018-7434-3 |___location=Baltimore, Maryland, USA |pages=48, 50 |url=https://books.google.com/books?id=7_0Thy4V3JIC&pg=PA65}}</ref>

<ref name="Goodall_1951">{{cite journal |author-first=William M. |author-last=Goodall |title=Television by Pulse Code Modulation |journal=[[Bell System Technical Journal]] |volume=30 |issue=1 |pages=33–49 |date=January 1951 |doi=10.1002/j.1538-7305.1951.tb01365.x}} (NB. Presented orally before the I.R.E. National Convention, New York City, March 1949.)</ref>

<ref name="Bhat-Savage_1996">{{cite journal |author-first1=Girish S. |author-last1=Bhat |author-first2=Carla Diane |author-last2=Savage |author-link2=Carla Diane Savage |title=Balanced Gray Codes |journal=[[Electronic Journal of Combinatorics]] |date=1996 |volume=3 |url=https://www.combinatorics.org/ojs/index.php/eljc/article/view/v3i1r25/ |issue=1 |article-number=R25 |doi=10.37236/1249 |doi-access=free}}</ref>

<ref name="Donohue_2003">{{cite web |author-first=Ryan |author-last=Donohue |title=Synchronization in Digital Logic Circuits |date=2003 |url=httphttps://www.stanford.edu/class/ee183/handouts_spr2003/synchronization_pres.pdf |access-date=2018-01-15 |url-status=live |archive-url=https://web.archive.org/web/20180115012747/https://web.stanford.edu/class/ee183/handouts_spr2003/synchronization_pres.pdf |archive-date=2018-01-15}}</ref>

<ref name="Mehta-Owens-Irwin_1996">{{cite book |author-last1=Mehta |author-first1=Huzefa |author-last2=Owens |author-first2=Robert Michael |author-last3=Irwin |author-first3=Mary Jane "Janie" |chapter=Some issues in gray code addressing |title=Proceedings of the Sixth Great Lakes Symposium on VLSI |date=1996-03-22 |issn=1066-1395 |doi=10.1109/GLSV.1996.497616 |chapter-url=https://ieeexplore.ieee.org/document/497616 |publisher=[[IEEE Computer Society]] |isbn=978-0-8186-7502-7 |pages=178–181|s2cid=52837310 }}</ref>

<ref name="Su-Tsui-Despain_1994">{{cite report |title=Low Power Architecture Design and Compilation Techniques for High-Performance Processors |author-first1=Ching-Long |author-last1=Su |author-first2=Chi-Ying |author-last2=Tsui |author-first3=Alvin M. |author-last3=Despain |date=1994 |publisher=Advanced Computer Architecture Laboratory |id=ACAL-TR-94-01 |url=http://www.scarpaz.com/2100-papers/Power%20Estimation/su94-low%20power%20architecture%20and%20compilation.pdf |access-date=2020-12-17 |url-status=live |archive-url=https://web.archive.org/web/20200726170626/http://www.scarpaz.com/2100-papers/Power%20Estimation/su94-low%20power%20architecture%20and%20compilation.pdf |archive-date=2020-07-26}}</ref>

<ref name="Guo-Parameswaran_2010">{{cite journal |author-first1=Hui |author-last1=Guo |author-first2=Sri |author-last2=Parameswaran |doi=10.1016/j.sysarc.2010.03.003 |volume=56 |issue=4–6 |date=April–June 2010 |title=Shifted Gray encoding to reduce instruction memory address bus switching for low-power embedded systems |journal=Journal of Systems Architecture |pages=180–190}}</ref>

<ref name="Dietz_2002">{{cite web |author-first=Henry Gordon "Hank" |author-last=Dietz |title=The Aggregate Magic Algorithms: Gray Code Conversion |work=The Aggregate |publisher=Electrical and Computer Engineering Department, College of Engineering, [[University of Kentucky]] |date=2002 |url=httphttps://aggregate.org/MAGIC/#Gray%20Code%20Conversion |access-date=2020-12-16 |url-status=live |archive-url=https://web.archive.org/web/20201216122620/http://aggregate.org/MAGIC/#Gray%2520Code%2520Conversion |archive-date=2020-12-16}}</ref>

<ref name="Guan_1998">{{cite journal |title=Generalized Gray Codes with Applications |author-last=Guan |author-first=Dah-Jyh |journal=Proceedings of the National Scientific Council, Republic of China, Part A |volume=22 |date=1998 |pages=841–848 |citeseerx=10.1.1.119.1344}}</ref>

<ref name="Suparta_2005">{{cite journal |author-first=I. Nengah |author-last=Suparta |title=A simple proof for the existence of exponentially balanced Gray codes |journal=[[Electronic Journal of Combinatorics]] |date=2005 |volume=12 |article-number=N19 |doi-access=free |doi=10.37236/1986}}</ref>

<ref name="Flahive-Bose_2007">{{cite journal |author-first1=Mary Elizabeth |author-last1=Flahive |author-link1=Mary Elizabeth Flahive |author-first2=Bella |author-last2=Bose |title=Balancing cyclic ''R''-ary Gray codes |journal=[[Electronic Journal of Combinatorics]] |date=2007 |volume=14 |article-number=R31 |doi-access=free |doi=10.37236/949}}</ref>

<ref name="Strackx-Piessens_2016">{{cite journal |author-first1=Raoul |author-last1=Strackx |author-first2=Frank |author-last2=Piessens |title=Ariadne: A Minimal Approach to State Continuity |journal=Usenix Security |date=2016 |volume=25 |url=https://distrinet.cs.kuleuven.be/software/sce/ariadne.html}}</ref>

<ref name="Savage-Winkler_1995">{{cite journal |author-first1=Carla Diane |author-last1=Savage |author-link1=Carla Diane Savage |author-first2=Peter |author-last2=Winkler |author-link2=Peter Winkler |title=Monotone Gray codes and the middle levels problem |journal=[[Journal of Combinatorial Theory]] | series=Series A |date=1995 |volume=70 |issn=0097-3165 |pages=230–248 |issue=2 |doi=10.1016/0097-3165(95)90091-8|doi-access=free}}</ref>

<ref name="Goddyn_1999">{{cite web |title=MATH 343 Applied Discrete Math Supplementary Materials |author-last=Goddyn |author-first=Luis |date=1999 |publisher=Department of Mathematics, [[Simon Fraser University]] |url=httphttps://www.math.sfu.ca/~goddyn/Courses/343/supMaterials.pdf |archive-url=https://web.archive.org/web/20150217160033/http://people.math.sfu.ca/~goddyn/Courses/343/supMaterials.pdf |archive-date=2015-02-17}}</ref>

<ref name="Sawada-Wong_2007">{{cite journal |author-first1=Joseph "Joe" |author-last1=Sawada |author-first2=Dennis Chi-Him |author-last2=Wong |title=A Fast Algorithm to generate Beckett–Gray codes |journal=Electronic Notes in Discrete Mathematics |volume=29 |pages=571–577 |date=2007 |doi=10.1016/j.endm.2007.07.091}}</ref>

<ref name="Spedding_1994_1">{{cite patent |inventor-last=Spedding |inventor-first=Norman Bruce<!-- Industrial Research Limited --> |pubdate=1994-10-28 |title=A position encoder |country=NZ |number=264738}}{{failed verification|date=July 2015}}</ref>

<ref name="Spedding_1994_2">{{cite web <!-- descriptive title: -->|title=The following is a copy of the provisional patent filed on behalf of Industrial Research Limited on 1994-10-28 – NZ Patent 264738 |author-first=Norman Bruce |author-last=Spedding |publisher=Industrial Research Limited |date=1994-10-28 |id=NZ Patent 264738 |url=http://www.winzurf.co.nz/Single_Track_Grey_Code_Patent/Single_track_Grey_code_encoder_patent.pdf |access-date=2018-01-14 |url-status=live |archive-url=https://web.archive.org/web/20171029205005/http://www.winzurf.co.nz/Single_Track_Grey_Code_Patent/Single_track_Grey_code_encoder_patent.pdf |archive-date=2017-10-29}}</ref>

<ref name="Hiltgen-Paterson-Brandestini_1996">{{cite journal |title=Single-Track Gray Codes |author-last1=Hiltgen |author-first1=Alain P. |author-first2=Kenneth G. |author-last2=Paterson |author-first3=Marco |author-last3=Brandestini |journal=[[IEEE Transactions on Information Theory]] |volume=42 |issue=5 |date=September 1996 |pages=1555–1561 |doi=10.1109/18.532900 |zbl=857.94007 |url=https://ieeexplore.ieee.org/document/532900}}</ref>

<ref name="Hiltgen-Paterson_2001">{{cite journal |title=Single-Track Circuit Codes |author-last1=Hiltgen |author-first1=Alain P. |author-first2=Kenneth G. |author-last2=Paterson |journal=[[IEEE Transactions on Information Theory]] |volume=47 |issue=6 |date=September 2001 |pages=2587–2595 |doi=10.1109/18.945274 |bibcode=2001ITIT...47.2587H |citeseerx=10.1.1.10.8218 |url=http://www.hpl.hp.com/techreports/2000/HPL-2000-81.pdf |access-date=2018-01-15 |url-status=live |archive-url=https://web.archive.org/web/20180115013155/http://www.hpl.hp.com/techreports/2000/HPL-2000-81.pdf |archive-date=2018-01-15}}</ref>

<ref name="Etzion-Schwartz_1999">{{cite journal |title=The Structure of Single-Track Gray Codes |author-last1=Etzion |author-first1=Tuvi |author-first2=Moshe |author-last2=Schwartz |journal=[[IEEE Transactions on Information Theory]] |volume=IT-45 |issue=7 |date=November 1999 |orig-date=1998-05-17 |pages=2383–2396 |doi=10.1109/18.796379 |citeseerx=10.1.1.14.8333 |url=http://etzion.net.technion.ac.il/files/2016/02/P54.pdf |access-date=2018-01-15 |url-status=live |archive-url=https://web.archive.org/web/20180115022531/http://etzion.net.technion.ac.il/files/2016/02/P54.pdf |archive-date=2018-01-15 }} [https://www.cs.technion.ac.il/users/wwwb/cgi-bin/tr-info.cgi?1998/CS/CS0937 Technical Report CS0937 <!-- https://web.archive.org/web/20180115023816/https://www.cs.technion.ac.il/users/wwwb/cgi-bin/tr-info.cgi?1998/CS/CS0937 -->] {{Webarchive|url=https://web.archive.org/web/20181215171438/http://www.cs.technion.ac.il/users/wwwb/cgi-bin/tr-info.cgi?1998/CS/CS0937 |date=2018-12-15 }}</ref>

<ref name="Sillke_1997">{{cite web |author-first=Torsten |author-last=Sillke |date=1997 |orig-date=1993-03-01 |title=Gray-Codes with few tracks (a question of Marco Brandestini) |url=http://www.mathematik.uni-bielefeld.de/~sillke/PROBLEMS/gray |access-date=2017-10-29 |url-status=live |archive-url=https://web.archive.org/web/20171029202303/https://www.math.uni-bielefeld.de/~sillke/PROBLEMS/gray |archive-date=2017-10-29}}</ref>

<ref name="Etzion-Paterson_1996">{{cite journal |title=Near Optimal Single-Track Gray Codes |author-first1=Tuvi |author-last1=Etzion |author-first2=Kenneth G. |author-last2=Paterson |journal=[[IEEE Transactions on Information Theory]] |volume=IT-42 |issue=3 |pages=779–789 |date=May 1996 |doi=10.1109/18.490544 |citeseerx=10.1.1.14.1527 |url=http://etzion.net.technion.ac.il/files/2016/02/P36.pdf |access-date=2018-04-08 |url-status=live |archive-url=https://web.archive.org/web/20161030214251/http://etzion.net.technion.ac.il/files/2016/02/P36.pdf |archive-date=2016-10-30}}</ref>

<ref name="Ruskey_2005">{{cite journal |title=A Survey of Venn Diagrams: Symmetric Diagrams |url=httphttps://www.combinatorics.org/Surveys/ds5/VennSymmEJC.html |journal=[[Electronic Journal of Combinatorics]] |date=2005-06-18 |author-first1=Frank |author-last1=Ruskey |author-link1=Frank Ruskey |author-first2=Mark |author-last2=Weston |doi=10.37236/26 |doi-access=free |department=Dynamic Surveys}}</ref>

<ref name="Alciatore-Histand_1999">{{cite book |author-last1=Alciatore |author-first1=David G. |author-first2=Michael B. |author-last2=Histand |title=Mechatronics |date=1999 |publisher=[[McGraw–Hill Education]] – Europe |isbn=978-0-07-131444-2 |url=httphttps://mechatronics.colostate.edu/}}</ref>

<ref name="Krishna_2008">{{cite web |author=Krishna |title=Gray code for QAM |date=2008-05-11 |url=httphttps://www.dsprelated.com/showthread/comp.dsp/96917-1.php |access-date=2017-10-29 |url-status=live |archive-url=https://web.archive.org/web/20171029192539/https://www.dsprelated.com/showthread/comp.dsp/96917-1.php |archive-date=2017-10-29}}</ref>

<ref name="Greferath_2009">{{cite book |editor-first1=Massimiliano |editor-last1=Sala |editor-first2=Teo |editor-last2=Mora |editor-first3=Ludovic |editor-last3=Perret |editor-first4=Shojiro |editor-last4=Sakata |editor-first5=Carlo |editor-last5=Traverso |title=Gröbner Bases, Coding, and Cryptography |url=https://archive.org/details/grbnerbasescodin00sala |url-access=limited |date=2009 |publisher=[[Springer Science & Business Media]] |isbn=978-3-540-93806-4 |chapter=An Introduction to Ring-Linear Coding Theory |author-first=Marcus |author-last=Greferath |page=[https://archive.org/details/grbnerbasescodin00sala/page/n226 220]}}</ref>

<ref name="Hazewinkel-Sole_2016">{{cite book |chapter=Kerdock and Preparata codes |author-first=Patrick |author-last=Solé |title=[[Encyclopedia of Mathematics]] |editor-first=Michiel |editor-last=Hazewinkel |editor-link=Michiel Hazewinkel |publisher=[[Springer Science+Business Media]] |date=2016 |isbn=978-1-4020-0609-8 |chapter-url=https://www.encyclopediaofmath.org/index.php/Kerdock_and_Preparata_codes |url-status=live |archive-url=https://web.archive.org/web/20171029191032/https://www.encyclopediaofmath.org/index.php/Kerdock_and_Preparata_codes |archive-date=2017-10-29}}</ref>

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* {{cite web |title=Gray Code Fundamentals |work=Design How-To |at=Part 1 |author-first=Clive "Max" |author-last=Maxfield |date=2012-10-01 |orig-date=2011-05-28 |publisher=[[EETimes]] |url=https://www.eetimes.com/document.asp?doc_id=1278809 |access-date=2017-10-30 |url-status=live |archive-url=https://web.archive.org/web/20171030135842/https://www.eetimes.com/document.asp?doc_id=1278809 |archive-date=2017-10-30}} [https://www.eetimes.com/document.asp?doc_id=1278827<!-- https://web.archive.org/web/20171030140209/https://www.eetimes.com/document.asp?doc_id=1278827 --> Part 2] [https://www.eetimes.com/document.asp?doc_id=1278853<!-- https://web.archive.org/web/20171030140323/https://www.eetimes.com/document.asp?doc_id=1278853 --> Part 3]

* {{cite book |title=Hacker's Delight |title-link=Hacker's Delight |chapter=Chapter 13: Gray Code |author-first=Henry S. |author-last=Warren, Jr. |date=2013 |edition=2 |publisher=[[Addison Wesley]] – [[Pearson Education, Inc.]] |isbn=978-0-321-84268-8 |pages=311–317<!-- page 318 is blank -->}} (7 pages)

* {{cite journal |title=Computing Binary Combinatorial Gray Codes Via Exhaustive Search With SAT Solvers |author-last1=Zinovik |author-first1=Igor |author-last2=Kroening |author-first2=Daniel |author-last3=Chebiryak |author-first3=Yury |journal=[[IEEE Transactions on Information Theory]] |publisher=[[IEEE]] |volume=54 |issue=4 |date=2008-03-21 |pages=1819–1823 |doi=10.1109/TIT.2008.917695 |urlbibcode=https://ieeexplore2008ITIT.ieee.org/document/4475394.54.1819Z |hdl=20.500.11850/11304 |s2cid=2854180 |hdl-access=free}} (5 pages)

* {{cite journal |author-first=Joseph A. |author-last=O'Brien |title=Unit-Distance Binary-Decimal Code Translators |journal=[[IRE Transactions on Electronic Computers]] |volume=EC-6 |issue=2 |date=June 1957 |issn=0367-9950 |doi=10.1109/TEC.1957.5221585 |pages=122–123 |url=https://www.researchgate.net/publication/250929745 |access-date=2020-05-25}} (2 pages)

* {{cite magazine |title=A decimal Gray code – Easily converted for shaft position coding |author-first=K. G. |author-last=Barr |___location=Faculty of Natural Sciences, [[University of the West Indies]] |date=March 1981 |magazine=[[Wireless World]] |volume=87 |number=1542 |pages=86–87 |url=https://worldradiohistory.com/hd2/IDX-Site-Early-Radio/Archive-Wireless-World-IDX/80s/Wireless-World-1981-03-OCR-Page-0046.pdf |access-date=2020-07-28 |url-status=live |archive-url=https://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 |archive-date=2020-07-28}}