Elias Gamma Encoding

Elias's gamma encoding is a universal code with a prefix. The prefix indicates the length of the binary string that follows it. It is therefore possible to encode any binary integer number.

To code a decimal number $ N $, take its binary representation $ N_{(2)} $ and calculate $ L = \lfloor \log_2 N \rfloor $ which is equivalent to its number of bits minus 1.

Example: To encode $ N = 5 $ which has for binary representation $ N_{(2)} = 101 $ (3 bits), calculate the integer part of $ \log_2 5 \approx 2.32 $ i.e. $ L = 2 $ (i.e. 1 less than the number of bits).

Encode in unary $ L $ and concatenate the binary representation without the most significant bit (the first 1) to obtain the Elias Gamma code $ \gamma $ that corresponds to $ N $

Example: $ L = 2 $ is coded in unary $ 001 $ (or sometimes $ 110 $) and the binary without the first $ 1 $ is $ 01 $ so the Elias encoding $ \gamma = 00101 $

Example: The first integers encoded with Gamma are&:

Read the binary value in order to split it into 2 sub-values of the same length separated by a bit: the first is the unary value corresponding to the number of bits minus 1, the second is the binary representation without the most significant bit.

To find the initial value, put a 1 in front of the second value and convert the binary number obtained to base 10.

Example: 00101 splits 00/1/01, the binary value is $ 101 $ which corresponds to the number $ 5 $.

Binary numbers have always an odd-length.

Coded values always start with a long string of 0 (or 1)

Elias Gamma encoding is generally linked to data compression algorithms.

The unary code can be coded with 0 followed by a separator 1 or else with 1 followed by the separator 0.

Example: 5 can be coded as 000001 or 111110.

Peter Elias described it in an article titled Universal codeword sets and representations of the integers in 1975.