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{{distinguish|Golomb coding}}
An '''exponential-Golomb code''' (or just '''Exp-Golomb code''') is a type of [[Universal code (data compression)|universal code]]. To encode any [[nonnegative integer]] ''x'' using the exp-Golomb code:
# Write down ''x''+1 in binary
# Count the bits written, subtract one, and write that number of starting zero bits preceding the previous bit string.
The first few values of the code
0
1
2
3
4
5
6
7
8
...<ref name="richardson"/>
In the above examples, consider the case 3. For 3, x+1 = 3 + 1 = 4. 4 in binary is '100'. '100' has 3 bits, and 3-1 = 2. Hence add 2 zeros before '100', which is '00100'
Similarly, consider 8. '8 + 1' in binary is '1001'. '1001' has 4 bits, and 4-1 is 3. Hence add 3 zeros before 1001, which is '0001001'.
This is identical to the [[Elias gamma code]] of ''x''+1, allowing it to encode 0.<ref>{{cite book |last = Rupp |first = Markus |title = Video and Multimedia Transmissions over Cellular Networks: Analysis, Modelling and Optimization in Live 3G Mobile Networks |year = 2009 |publisher = Wiley |pages = 149 |isbn = 9780470747766 |url = https://books.google.com/books?id=H9hUBT-JvUoC&q=Exponential-Golomb+coding&pg=PA149 }}</ref>
==Extension to negative numbers==
Exp-Golomb coding is used in the [[H.264/MPEG-4 AVC]] and H.265 [[High Efficiency Video Coding]] video compression standards, in which there is also a variation for the coding of signed numbers by assigning the value 0 to the binary codeword '0' and assigning subsequent codewords to input values of increasing magnitude (and alternating sign, if the field can contain a negative number):
0 ⇒ 0 ⇒ 1 ⇒ 1
1 ⇒ 1 ⇒ 10 ⇒ 010
−1 ⇒ 2 ⇒ 11 ⇒ 011
2 ⇒ 3 ⇒ 100 ⇒ 00100
−2 ⇒ 4 ⇒ 101 ⇒ 00101
3 ⇒ 5 ⇒ 110 ⇒ 00110
−3 ⇒ 6 ⇒ 111 ⇒ 00111
4 ⇒ 7 ⇒ 1000 ⇒ 0001000
−4 ⇒ 8 ⇒ 1001 ⇒ 0001001
...<ref name="richardson">{{cite book |last = Richardson |first = Iain |title = The H.264 Advanced Video Compression Standard |year = 2010 |publisher = Wiley |isbn = 978-0-470-51692-8 |pages = 208, 221 |url = https://books.google.com/books?id=LJoDiPnBzQ8C&q=Exponential-Golomb+coding&pg=PA221 }}</ref>
In other words, a non-positive integer ''x''≤0 is mapped to an even integer −2''x'', while a positive integer ''x''>0 is mapped to an odd integer 2''x''−1.
Exp-Golomb coding is also used in the [[Dirac (video compression format)|Dirac video codec]].<ref>{{cite web |title = Dirac Specification |url = http://diracvideo.org/download/specification/dirac-spec-latest.pdf |publisher = BBC |access-date = 9 March 2011 |url-status = usurped |archive-url = https://web.archive.org/web/20150503015104/http://diracvideo.org/download/specification/dirac-spec-latest.pdf |archive-date = 2015-05-03 }}</ref>
==Generalization to order ''k''==
To encode larger numbers in fewer bits (at the expense of using more bits to encode smaller numbers), this can be generalized using a [[nonnegative integer]] parameter ''k''. To encode a nonnegative integer ''x'' in an order-''k'' exp-Golomb code:
# Encode ⌊''x''/2<sup>''k''</sup>⌋ using order-0 exp-Golomb code described above, then
# Encode ''x'' mod 2<sup>''k''</sup> in binary with k bits
An equivalent way of expressing this is:
# Encode ''x''+2<sup>''k''</sup>−1 using the order-0 exp-Golomb code (i.e. encode ''x''+2<sup>''k''</sup> using the Elias gamma code), then
# Delete ''k'' leading zero bits from the encoding result
{|class=wikitable
|+ Exp-Golomb-''k'' coding examples
! ''x'' || ''k''=0 || ''k''=1 || ''k''=2 ||''k''=3
|rowspan=11|
! ''x'' || ''k''=0 || ''k''=1 || ''k''=2 ||''k''=3
|rowspan=11|
! ''x'' || ''k''=0 || ''k''=1 || ''k''=2 ||''k''=3
|-
| 0 || 1 || 10 || 100 || 1000
| 10 || 0001011 || 001100 || 01110 || 010010
| 20 || 000010101 || 00010110 || 0011000 || 011100
|-
| 1 || 010 || 11 || 101 || 1001
| 11 || 0001100 || 001101 || 01111 || 010011
| 21 || 000010110 || 00010111 || 0011001 || 011101
|-
| 2 || 011 || 0100 || 110 || 1010
| 12 || 0001101 || 001110 || 0010000 || 010100
| 22 || 000010111 || 00011000 || 0011010 || 011110
|-
| 3 || 00100 || 0101 || 111 || 1011
| 13 || 0001110 || 001111 || 0010001 || 010101
| 23 || 000011000 || 00011001 || 0011011 || 011111
|-
| 4 || 00101 || 0110 || 01000 || 1100
| 14 || 0001111 || 00010000 || 0010010 || 010110
| 24 || 000011001 || 00011010 || 0011100 || 00100000
|-
| 5 || 00110 || 0111 || 01001 || 1101
| 15 || 000010000 || 00010001 || 0010011 || 010111
| 25 || 000011010 || 00011011 || 0011101 || 00100001
|-
| 6 || 00111 || 001000 || 01010 || 1110
| 16 || 000010001 || 00010010 || 0010100 || 011000
| 26 || 000011011 || 00011100 || 0011110 || 00100010
|-
| 7 || 0001000 || 001001 || 01011 || 1111
| 17 || 000010010 || 00010011 || 0010101 || 011001
| 27 || 000011100 || 00011101 || 0011111 || 00100011
|-
| 8 || 0001001 || 001010 || 01100 || 010000
| 18 || 000010011 || 00010100 || 0010110 || 011010
| 28 || 000011101 || 00011110 || 000100000 || 00100100
|-
| 9 || 0001010 || 001011 || 01101 || 010001
| 19 || 000010100 || 00010101 || 0010111 || 011011
| 29 || 000011110 || 00011111 || 000100001 || 00100101
|}
==See also==
*[[Elias gamma coding|Elias gamma (γ) coding]]
*[[Elias delta coding|Elias delta (δ) coding]]
*[[Elias omega coding|Elias omega (ω) coding]]
*[[Universal code (data compression)|Universal code]]
==References==
{{Reflist}}
{{Compression Methods}}
{{DEFAULTSORT:Exponential-Golomb Coding}}
[[Category:Entropy coding]]
[[Category:Numeral systems]]
[[Category:Data compression]]
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