Elias gamma code is a universal code encoding positive integers developed by Peter Elias. It is used most commonly when coding integers whose upperbound cannot be determined beforehand.
Contents
Encoding
To code a number:
An equivalent way to express the same process:
The code begins (the implied probability distribution for the code is added for clarity):
Decoding
To decode an Elias gammacoded integer:
Uses
Gamma coding is used in applications where the largest encoded value is not known ahead of time, or to compress data in which small values are much more frequent than large values.
Generalizations
Gamma coding does not code zero or negative integers. One way of handling zero is to add 1 before coding and then subtract 1 after decoding. Another way is to prefix each nonzero code with a 1 and then code zero as a single 0. One way to code all integers is to set up a bijection, mapping integers (0, 1, 1, 2, 2, 3, 3, ...) to (1, 2, 3, 4, 5, 6, 7, ...) before coding.
ExponentialGolomb coding generalizes the gamma code to integers with a "flatter" powerlaw distribution, just as Golomb coding generalizes the unary code. It involves dividing the number by a positive divisor, commonly a power of 2, writing the gamma code for one more than the quotient, and writing out the remainder in an ordinary binary code.
Example code
Encode
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