Prefix Code

A prefix code is a type of code system (typically a variable-length code) distinguished by its possession of the "prefix property"; which states that there is no valid code word in the system that is a prefix (start) of any other valid code word in the set. For example, a code with code words {9, 59, 55} has the prefix property; a code consisting of {9, 5, 59, 55} does not, because "5" is a prefix of both "59" and "55". With a prefix code, a receiver can identify each word without requiring a special marker between words.

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 but in most mathematical books and articles (e. g.) it is used to mean self-synchronizing codes, 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 markers to 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, for example {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 codes, country calling codes, the country and publisher parts of ISBNs, the Secondary Synchronization Codes used in the UMTS W-CDMA 3G Wireless Standard, and the instruction sets (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.

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