Elias${\displaystyle \gamma }$ code orElias gamma code is auniversal code encoding positive integers developed byPeter Elias.^{[1]: 197, 199} It is used most commonly when coding integers whose upper bound cannot be determined beforehand.

To code anumberx ≥ 1:

1. Let${\displaystyle N=\lfloor {\log }_{2}x\rfloor }$ be the highest power of 2 it contains, so 2^N ≤x < 2^N+1.
2. Write out${\displaystyle N}$ zero bits, then
3. Append thebinary form of${\displaystyle x}$, an${\displaystyle (N+1)}$-bit binary number.

An equivalent way to express the same process:

1. Encode${\displaystyle N}$ inunary; that is, as${\displaystyle N}$ zeroes followed by a one.
2. Append the remaining${\displaystyle N}$ binary digits of${\displaystyle x}$ to this representation of${\displaystyle N}$.

To represent a number${\displaystyle x}$, Elias gamma (γ) uses${\displaystyle 2\lfloor {\log }_2(x)\rfloor +1}$ bits.^[1]: 199

The code begins (theimplied probability distribution for the code is added for clarity):

To decode an Elias gamma-coded integer:

1. Read and count 0s from the stream until you reach the first 1. Call this count of zeroesN.
2. Considering the one that was reached to be the first digit of the integer, with a value of 2^N, read the remainingN digits of the integer.

Gamma coding is used in applications where the largest encoded value is not known ahead of time, or tocompress data in which small values are much more frequent than large values.

Gamma coding can be more size efficient in those situations. For example, note that, in the table above, if a fixed 8-bit size is chosen to store a small number like the number 5, the resulting binary would be00000101, while the γ-encoding variable-bit version would be00 1 01, needing 3 bits less. On the contrary, bigger values, like 254 stored in fixed 8-bit size, would be11111110 while the γ-encoding variable-bit version would be0000000 1 1111110, needing 7 extra bits.

Gamma coding is a building block in theElias delta code.

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 abijection, mapping integers (0, −1, 1, −2, 2, −3, 3, ...) to (1, 2, 3, 4, 5, 6, 7, ...) before coding. In software, this is most easily done by mapping non-negative inputs to odd outputs, and negative inputs to even outputs, so the least-significant bit becomes an invertedsign bit:

Exponential-Golomb coding generalizes the gamma code to integers with a "flatter" power-law distribution, just asGolomb 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.

• Elias delta (δ) coding – Universal code encoding positive integers
• Elias omega (ω) coding – Universal code encoding positive integers
• Posit (number format) – Variant of floating-point numbers in computersPages displaying short descriptions of redirect targets

• Sayood, Khalid (2003). "Levenstein and Elias Gamma Codes".Lossless Compression Handbook.Elsevier.ISBN 978-0-12-620861-0.

• Entropy coding
• Numeral systems
• Data compression

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