Prefix code: Difference between revisions

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{{Short description|Type of code system}}

<ref name=":0" />A '''prefix code''' is a type of [[code]] system distinguished by its possession of the "'''prefix property"''', which requires that there is no whole [[Code word (communication)|code word]] in the system that is a [[prefix (computer science)|prefix]] (initial segment) of any other code word in the system. It is trivially true for fixed-length codecodes, so only a point of consideration infor [[variable-length code|variable-length codes]].

For example, a code with code words {9,&nbsp;55} has the prefix property; a code consisting of {9,&nbsp;5,&nbsp;59,&nbsp;55} does not, because "5" is a prefix of "59" and also of "55". A prefix code is a [[uniquely decodable code]]: given a complete and accurate sequence, a receiver can identify each word without requiring a special marker between words. However, there are uniquely decodable codes that are not prefix codes; for instance, the reverse of a prefix code is still uniquely decodable (it is a suffix code), but it is not necessarily a prefix code.

Prefix codes are also known as '''prefix-free codes''', '''prefix condition codes''' and '''instantaneous codes'''. Although [[Huffman coding]] is just one of many algorithms for deriving prefix codes, prefix codes are also widely referred to as "Huffman codes", even when the code was not produced by a Huffman algorithm. The term '''comma-free code''' is sometimes also applied as a synonym for prefix-free codes<ref>US [[Federal Standard 1037C]]</ref><ref name=":0">{{citation|title=ATIS Telecom Glossary 2007|url=http://www.atis.org/glossary/definition.aspx?id=6416|accessdateaccess-date=December 4, 2010|archive-date=July 8, 2010|archive-url=https://web.archive.org/web/20100708083829/http://www.atis.org/glossary/definition.aspx?id=6416|url-status=dead}}</ref> but in most mathematical books and articles (e.g.<ref>{{citation|last1=Berstel|first1=Jean|last2=Perrin|first2=Dominique|title=Theory of Codes|publisher=Academic Press|year=1985}}</ref><ref>{{citation|doi=10.4153/CJM-1958-023-9|last1=Golomb|first1=S. W.|author1-link=Solomon W. Golomb|last2=Gordon|first2=Basil|author2-link=Basil Gordon|last3=Welch|first3=L. R.|title=Comma-Free Codes|journal=Canadian Journal of Mathematics|volume=10|issue=2|pages=202–209|year=1958|s2cid=124092269 |url=https://books.google.com/books?id=oRgtS14oa-sC&pg=PA202|doi-access=free}}</ref>) a comma-free code is used to mean a [[self-synchronizing code]], a subclass of prefix codes.

Using prefix codes, a message can be transmitted as a sequence of concatenated code words, without any [[Out-of-band data|out-of-band]] markers or (alternatively) special markers between words to [[framing (telecommunication)|frame]] the words in the message. The recipient can decode the message unambiguously, by repeatedly finding and removing sequences that form valid code words. This is not generally possible with codes that lack the prefix property, for example {0,&nbsp;1,&nbsp;10,&nbsp;11}: a receiver reading a "1" at the start of a code word would not know whether that was the complete code word "1", or merely the prefix of the code word "10" or "11"; so the string "10" could be interpreted either as a single codeword or as the concatenation of the words "1" then "0".

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[[Huffman coding]] is a more sophisticated technique for constructing variable-length prefix codes. The Huffman coding algorithm takes as input the frequencies that the code words should have, and constructs a prefix code that minimizes the weighted average of the code word lengths. (This is closely related to minimizing the entropy.) This is a form of [[lossless data compression]] based on [[entropy encoding]].

Some codes mark the end of a code word with a special "comma" symbol (also called a [[Sentinel value]]), different from normal data.<ref>[{{cite web |url=http://www.imperial.ac.uk/research/hep/group/theses/JJones.pdf "|title=Development of Trigger and Control Systems for CMS"] by |first1=J. |last1=A. Jones: "Synchronisation"|page=70 p|publisher=High Energy Physics, Blackett Laboratory, Imperial College, London |url-status=dead |archive-url= https://web.archive.org/web/20110613183447/http://www.imperial.ac.uk/research/hep/group/theses/JJones.pdf 70|archive-date= Jun 13, 2011 }}</ref> This is somewhat analogous to the spaces between words in a sentence; they mark where one word ends and another begins. If every code word ends in a comma, and the comma does not appear elsewhere in a code word, the code is automatically prefix-free. However, modernreserving communicationan systemsentire sendsymbol everythingonly asfor sequencesuse of "1" and "0"&nbsp;– addingas a thirdcomma symbol wouldcan be expensiveinefficient, andespecially usingfor itlanguages onlywith ata thesmall endsnumber of words would be inefficientsymbols. [[Morse code]] is an everyday example of a variable-length code with a comma. The long pauses between letters, and the even longer pauses between words, help people recognize where one letter (or word) ends, and the next begins. Similarly, [[Fibonacci coding]] uses a "11" to mark the end of every code word.

[[Self-synchronizing code]]s are prefix codes that allow [[frame synchronization]].

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* [[Golomb Rice code]]

* [[Straddling checkerboard]] (simple cryptography technique which produces prefix codes)

* [[Vbinarybinary coding]]<ref>{{citation|doi=10.25209/2079-3316-2018-9-4-239-252|last1=Shevchuk|first1=Y. V.|author1-link=Yury V. Shevchuk|title=Vbinary: variable length integer coding revisited|journal=Program Systems: Theory and Applications|volume=9|issue=4|pages=239-252239–252|year=2018|url=http://psta.psiras.ru//read/psta2018_4_239-252.pdf|doi-access=free}}</ref>

==Notes==

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==References==

* {{cite book | last1=Berstel | first1=Jean | last2=Perrin | first2=Dominique | last3=Reutenauer | first3=Christophe | title=Codes and automata | series=Encyclopedia of Mathematics and its Applications | volume=129 | ___location=Cambridge | publisher=[[Cambridge University Press]] | year=2010 | url=http://www-igm.univ-mlv.fr/~berstel/LivreCodes/Codes.html | isbn=978-0-521-88831-8 | zbl=1187.94001 }}

* {{cite journal | last=Elias | first=Peter | authorlinkauthor-link=Peter Elias | title=Universal codeword sets and representations of the integers | journal=IEEE Trans. Inf. Theory | volume=21 | number=2 | year=1975 | pages=194–203 | issn=0018-9448 | zbl=0298.94011 | doi=10.1109/tit.1975.1055349}}

* D.A. Huffman, "A method for the construction of minimum-redundancy codes", Proceedings of the I.R.E., Sept. 1952, pp.&nbsp;1098–1102 (Huffman's original article)

* [https://web.archive.org/web/20070220234037/http://www.huffmancoding.com/david/scientific.html Profile: David A. Huffman], [[Scientific American]], Sept. 1991, pp.&nbsp;54–58 (Background story)

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==External links==

* [http://plus.maths.org/issue10/features/infotheory/index.html Codes, trees and the prefix property] by Kona Macphee

{{Compression methods}}

[[Category:Coding theory]]

[[Category:Prefixes|code]]

[[Category:Data compression]]

[[Category:Lossless compression algorithms]] <!-- do I really need both categories? -->