Tool to convert numbers according to the Elias Gamma code, denoted γ, generally used in data compression to store data of arbitrary size.
Elias Gamma Encoding - dCode
Tag(s) : Compression, Notation System
dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day!
A suggestion ? a feedback ? a bug ? an idea ? Write to dCode!
Tool to convert numbers according to the Elias Gamma code, denoted γ, generally used in data compression to store data of arbitrary size.
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&:
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 | 18 | 000010010 |
19 | 000010011 | 20 | 000010100 |
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.
dCode retains ownership of the online 'Elias Gamma Encoding' tool source code. Except explicit open source licence (indicated CC / Creative Commons / free), any algorithm, applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or any function (convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (PHP, Java, C#, Python, Javascript, Matlab, etc.) no data, script or API access will be for free, same for Elias Gamma Encoding download for offline use on PC, tablet, iPhone or Android !
Please, check our community Discord for help requests!