Variable-length code

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.

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 ${\displaystyle S}$ and ${\displaystyle T}$ be two finite sets, called the source and target alphabets, respectively. A code ${\displaystyle C:S\to T^{*}}$ is a total function mapping each symbol from ${\displaystyle S}$ to a sequence of symbols over ${\displaystyle T}$, and the extension of ${\displaystyle C}$ to a homomorphism of ${\displaystyle S^{*}}$ into ${\displaystyle T^{*}}$, which naturally maps each sequence of source symbols to a sequence of target symbols, is referred to as its extension.

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