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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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[[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<sup>''i''</sup> on 2<sup>''i''</sup> off. The most significant digit is an exception to this: for an ''n''-bit Gray code, the most significant digit follows the pattern 2<sup>''n''-1</sup> on, 2<sup>''n''-1</sup> off, which is the same (cyclic) sequence of values as for the second-most significant digit, but shifted forwards 2<sup>''n''-2</sup> 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<sup>''i''</sup> &nbsp;on 2<sup>''i''</sup> &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<sup>''n''-1−1</sup> &nbsp;on, 2<sup>''n''-1−1</sup> &nbsp;off, which is the same (cyclic) sequence of values as for the second-most significant digit, but shifted forwards 2<sup>''n''-2−2</sup> places. The four-bit version of this is shown below:
 
{| class="wikitable sortable" style="text-align:center;"
|-
! Decimal !! Binary !! Gray
|-
Line 1,007 ⟶ 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 1,029 ⟶ 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 1,039 ⟶ 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 1,049 ⟶ 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 1,059 ⟶ 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 1,069 ⟶ 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 1,079 ⟶ 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 1,089 ⟶ 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 1,099 ⟶ 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 1,109 ⟶ 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 1,119 ⟶ 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 1,129 ⟶ 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 1,169 ⟶ 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 1,195 ⟶ 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 1,205 ⟶ 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 1,235 ⟶ 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}}
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