Variable-length code

This article is about the transmission of data across noisy channels. For the storage of text in computers, see Variable-width encoding.

In coding theory a variable-length code is a code which maps source symbols to a variable number of bits.

Variable-length codes can allow sources to be compressed and decompressed with zero error (lossless data compression) and still be read back symbol by symbol. With the right coding strategy an independent and identically-distributed source may be compressed almost arbitrarily close to its entropy. This is in contrast to fixed length coding methods, for which data compression is only possible for large blocks of data, and any compression beyond the logarithm of the total number of possibilities comes with a finite (though perhaps arbitrarily small) probability of failure.

Some examples of well-known variable-length coding strategies are Huffman coding, Lempel–Ziv coding and arithmetic coding.

Codes and their extensions

The extension of a code is the mapping of finite length source sequences to finite length bit strings, that is obtained by concatenating for each symbol of the source sequence the corresponding codeword produced by the original code.

Using terms from formal language theory, the precise mathematical definition is as follows: Let and be two finite sets, called the source and target alphabets, respectively. A code is a total function mapping each symbol from to a sequence of symbols over , and the extension of to a homomorphism of into , which naturally maps each sequence of source symbols to a sequence of target symbols, is referred to as its extension.

Classes of variable-length codes

Variable-length codes can be strictly nested in order of decreasing generality as non-singular codes, uniquely decodable codes and prefix codes. Prefix codes are always uniquely decodable, and these in turn are always non-singular:

Non-singular codes

A code is non-singular if each source symbol is mapped to a different non-empty bit string, i.e. the mapping from source symbols to bit strings is injective.

Uniquely decodable codes

A code is uniquely decodable if its extension is non-singular. Whether a given code is uniquely decodable can be decided with the Sardinas–Patterson algorithm.

Prefix codes

Main article: Prefix code

A code is a prefix code if no target bit string in the mapping is a prefix of the target bit string of a different source symbol in the same mapping. This means that symbols can be decoded instantaneously after their entire codeword is received. Other commonly used names for this concept are prefix-free code, instantaneous code, or context-free code.

Symbol Codeword
a 0
b 10
c 110
d 111
Example of encoding and decoding:
aabacdab → 00100110111010 → |0|0|10|0|110|111|0|10| → aabacdab

A special case of prefix codes are block codes. Here all codewords must have the same length. The latter are not very useful in the context of source coding, but often serve as error correcting codes in the context of channel coding.


The advantage of a variable-length code is that unlikely source symbols can be assigned longer codewords and likely source symbols can be assigned shorter codewords, thus giving a low expected codeword length. For the above example, if the probabilities of (a, b, c, d) were , the expected number of bits used to represent a source symbol using the code above would be:


As the entropy of this source is 1.7500 bits per symbol, this code compresses the source as much as possible so that the source can be recovered with zero error.


  1. Berstel et al. (2009), Example 2.3.1, p. 63


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