In coding theory, variable-length encoding is a symbol encoding scheme in which codes of differing lengths are used to encode symbols for representation through a communication channel or in a storage medium. The equivalent concept in computer science is bit string. 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. 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. For these reasons, they were sometimes used to pack English text into fewer bytes in adventure games for early microcomputers. However, disks, increases in computer memory, and general purpose compression algorithms have rendered such methods obsolete. Multibyte encodings are usually the result of a need to increase the number of characters which can be encoded without breaking backward compatibility with an existing constraint. For example, with one byte (8 bits) per character, one can encode 256 possible characters; in order to encode more than 256 characters, the obvious choice would be to use two or more bytes per encoding unit, two bytes (16 bits) would allow 65,536 possible characters, but such a change would break compatibility with existing systems and therefore might not be feasible at all. Unlikely source symbols can be assigned longer codewords while likely source symbols can be assigned shorter codewords, thus giving a low expected codeword length. Some examples of well-known variable-length coding strategies are Huffman coding, Lempel–Ziv coding, arithmetic coding, and context-adaptive variable-length coding.
General structure A multibyte encoding system minimises disruption to existing software by keeping some characters as single-unit codes, while others require multiple units. This creates three unit types: singletons (which consist of a single unit), lead units (which come first in a multiunit sequence), and trail units (which come afterwards in a multiunit sequence). Input and display systems must handle these structures, though most other software does not. For example, the four character string "I♥NY" is encoded in UTF-8 like this (shown as hexadecimal byte values): 49 E2 99 A5 4E 59. Of the six units in that sequence, 49, 4E, and 59 are singletons (for I, N, and Y), E2 is a lead unit and 99 and A5 are trail units. The heart symbol is represented by the combination of the lead unit and the two trail units. UTF-8 clearly distinguishes singletons, leads, and trails with non-overlapping value ranges. By contrast, older encodings often reuse values, making it harder to parse text correctly. This can cause false positives in searches or make a corrupted byte disrupt long sequences. In well-designed encodings like UTF-8, searching works reliably, and corruption affects only the character containing the bad unit.
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 S {\displaystyle S} and T {\displaystyle T} be two finite sets, called the source and target alphabets, respectively. A code C : S → T ∗ {\displaystyle C:S\to T^{*}} is a total function mapping each symbol from S {\displaystyle S} to a sequence of symbols over T {\displaystyle T} , and the extension of C {\displaystyle C} to a homomorphism of S ∗ {\displaystyle S^{*}} into T ∗ {\displaystyle T^{*}} , which naturally maps each sequence of source symbols to a sequence of target symbols, is referred to as its extension. 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:
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