Elias gamma coding

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Elias gamma code is a universal code encoding the positive integers. To code a number:

1. Write it in binary.
2. Subtract 1 from the number of bits written in step 1 and prepend that many zeros.

An equivalent way to express the same process:

1. Separate the integer into the highest power of 2 it contains (2^N) and the remaining N binary digits of the integer.
2. Encode N in unary; that is, as N zeroes followed by a one.
3. Append the remaining N binary digits to this representation of N.

The code begins:

 1 1
 2 010
 3 011
 4 00100
 5 00101
 6 00110
 7 00111
 8 0001000
 9 0001001
10 0001010
11 0001011
12 0001100
13 0001101
14 0001110
15 0001111
16 000010000
17 000010001

To decode an Elias gamma-coded integer:

1. Read and count zeroes from the stream until you reach the first one. Call this count of zeroes N.
2. Considering the one that was reached to be the first digit of the integer, with a value of 2^N, read the remaining N digits of the integer.

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 the 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.

See also

• Elias delta coding
• Elias omega coding