Prefix code: Difference between revisions

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Using prefix codes, a message can be transmitted as a sequence of concatenated code words, without any [[out-of-band]] markers to [[framing (telecommunication)|frame]] the words in the message. The recipient can decode the message unambiguously, by repeatedly finding and removing prefixes that form valid code words. This is not possible with codes that lack the prefix property, such as our example of {0, 1, 10, 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".

The variable-length [[Huffman coding|Huffman codes]], [[country calling codes]], the country and publisher parts of [[ISBN]]s, and the Secondary Synchronization Codes used in the [[UMTS]] [[W-CDMA]] 3G Wireless Standard are prefix codes. Prefix codes are also a form of [[entropy encoding]] used in [[lossless data compression]].

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

[[Kraft's inequality]] characterizes the sets of code word lengths that are possible in a prefix code.

== Techniques ==

Techniques for constructing a prefix code can be simple, or quite complicated.

If every word in the code has the same length, the code is called a '''fixed-length code''', or a '''block code''' (though the term [[block code]] is also used for fixed-lengthsize [[error-correcting code]]s in [[channel coding]]). For example, [[ISO 8859-15]] letters are always 8 bits long. [[UTF-32/UCS-4]] letters are always 32 bits long. [[Asynchronous Transfer Mode|ATM packets]] are always 424 bits long. A block code of fixed length ''k'' bits can encode up to <math>2^{k}</math> source symbols.

Prefixes cannot exist in a fixed-length code without padding fixed codes to the shorter prefixes in order to meet the length of the longest prefixes (however such padding codes may be selected to introduce redundancy that allows autocorrection and/or synchronisation). UnfortunatelyHowever, fixed length encodings are inefficient in situations where some words are much more likely to be transmitted than others (in which case some or all of the redundancy may be eliminated for data compression).

[[Truncated binary encoding]] is a straightforward generalization of block codes to deal with cases where the number of symbols ''n'' is not a power of two. Source symbols are assigned codewords of length ''k'' and ''k''+1. where <math>2^{k} < n < 2^{k+1}</math>.

Some codes mark the end of a code word with a special "comma" symbol, different from normal data.

<ref>

[http://www.imperial.ac.uk/research/hep/group/theses/JJones.pdf "Development of Trigger and Control Systems for CMS"] by J. A. Jones: "Synchronisation" p. 70

</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 prefix-free. However, modern communication systems send everything as sequences of "1" and "0" &ndash; adding a third symbol would be expensive, and using it only at the ends of words would be inefficient. [[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.

[[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 a form of [[lossless data compression]] based on [[entropy encoding]].

[[Kraft'sSome inequality]]codes characterizesmark the setsend of a code word lengthswith thata arespecial possible"comma" insymbol, adifferent prefixfrom codenormal data. <ref>

[http://www.imperial.ac.uk/research/hep/group/theses/JJones.pdf "Development of Trigger and Control Systems for CMS"] by J. A. Jones: "Synchronisation" p. 70

</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 prefix-free. However, modern communication systems send everything as sequences of "1" and "0" &ndash; adding a third symbol would be expensive, and using it only at the ends of words would be inefficient. [[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]].

==Prefix codes in use today==

Examples of prefix codes include:

* [[country calling codes]]