Editing Prefix code (section)

{{Short description|Type of code system}}
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 codes, so only a point of consideration for [[variable-length code|variable-length codes]].

For example, a code with code {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>{{citation|title=ATIS Telecom Glossary 2007|url=http://www.atis.org/glossary/definition.aspx?id=6416|access-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".

The variable-length [[Huffman coding|Huffman codes]], [[country calling codes]], the country and publisher parts of [[ISBN]]s, the Secondary Synchronization Codes used in the [[UMTS]] [[W-CDMA]] 3G Wireless Standard, and the [[instruction set]]s (machine language) of most computer microarchitectures are prefix codes.

Prefix codes are not [[error-correcting codes]]. In practice, a message might first be compressed with a prefix code, and then encoded again with [[channel coding]] (including error correction) before transmission.

For any [[Variable-length_code#Uniquely_decodable_codes|uniquely decodable]] code there is a prefix code that has the same code word lengths.<ref name=LTU2015>Le Boudec, Jean-Yves, Patrick Thiran, and Rüdiger Urbanke. Introduction aux sciences de l'information: entropie, compression, chiffrement et correction d'erreurs. PPUR Presses polytechniques, 2015.</ref> [[Kraft's inequality]] characterizes the sets of code word lengths that are possible in a [[Variable-length_code#Uniquely_decodable_codes|uniquely decodable]] code.<ref name=BRS75>Berstel et al (2010) p.75</ref>