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| {{redirect|Gamma encoding|the signal processing operation|gamma correction}}
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| '''Elias gamma code''' is a [[universal code (data compression)|universal code]] encoding positive integers developed by [[Peter Elias]]<ref name="Elias">{{cite journal | first = Peter | last = Elias | title = Universal codeword sets and representations of the integers | journal = [[IEEE Transactions on Information Theory]] | volume = 21 | issue = 2 | pages = 194–203 |date=March 1975 | url = http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=1055349}}</ref>{{rp|197, 199}}. It is used most commonly when coding integers whose upper-bound cannot be determined beforehand.
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| ==Encoding==
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| To code a [[number]]:
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| #Write it in [[binary numeral system|binary]].
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| #Subtract 1 from the number of bits written in step 1 and prepend that many zeros.
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| An equivalent way to express the same process:
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| #Separate the integer into the highest power of 2 it contains (2<sup>''N''</sup>) and the remaining ''N'' binary digits of the integer.
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| #Encode ''N'' in [[Unary numeral system|unary]]; that is, as ''N'' zeroes followed by a one.
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| #Append the remaining ''N'' binary digits to this representation of ''N''.
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| To represent a number <math>x</math>, Elias gamma uses <math>2 \lfloor \log_2(x) \rfloor + 1</math> bits<ref name="Elias"/>{{rp|199}}.
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| The code begins (the [[implied probability]] distribution for the code is added for clarity):
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| {| class=wikitable
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| ! Number !! Encoding !! Implied probability
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| |-
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| | 1 = 2<sup>0</sup> + ''0'' || <code>1</code> || 1/2
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| |-
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| | 2 = 2<sup>1</sup> + ''0'' || <code>01''0''</code> || 1/8
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| |-
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| | 3 = 2<sup>1</sup> + ''1'' || <code>01''1''</code> || 1/8
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| |-
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| | 4 = 2<sup>2</sup> + ''0'' || <code>001''00''</code> || 1/32
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| |-
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| | 5 = 2<sup>2</sup> + ''1'' || <code>001''01''</code> || 1/32
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| |-
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| | 6 = 2<sup>2</sup> + ''2'' || <code>001''10''</code> || 1/32
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| |-
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| | 7 = 2<sup>2</sup> + ''3'' || <code>001''11''</code> || 1/32
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| |-
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| | 8 = 2<sup>3</sup> + ''0'' || <code>0001''000''</code> || 1/128
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| |-
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| | 9 = 2<sup>3</sup> + ''1'' || <code>0001''001''</code> || 1/128
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| |-
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| | 10 = 2<sup>3</sup> + ''2'' || <code>0001''010''</code> || 1/128
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| |-
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| | 11 = 2<sup>3</sup> + ''3'' || <code>0001''011''</code> || 1/128
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| |-
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| | 12 = 2<sup>3</sup> + ''4'' || <code>0001''100''</code> || 1/128
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| |-
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| | 13 = 2<sup>3</sup> + ''5'' || <code>0001''101''</code> || 1/128
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| |-
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| | 14 = 2<sup>3</sup> + ''6'' || <code>0001''110''</code> || 1/128
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| |-
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| | 15 = 2<sup>3</sup> + ''7'' || <code>0001''111''</code> || 1/128
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| |-
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| | 16 = 2<sup>4</sup> + ''0'' || <code>00001''0000''</code> || 1/512
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| |-
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| | 17 = 2<sup>4</sup> + ''1'' || <code>00001''0001''</code> || 1/512
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| |}
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| ==Decoding==
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| To decode an Elias gamma-coded integer:
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| #Read and count 0s from the stream until you reach the first 1. Call this count of zeroes ''N''.
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| #Considering the one that was reached to be the first digit of the integer, with a value of 2<sup>''N''</sup>, read the remaining ''N'' digits of the integer.
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| ==Uses==
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| Gamma coding is used in applications where the largest encoded value is not known ahead of time, or to [[Data compression|compress]] data in which small values are much more frequent than large values.
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| == Generalizations ==<!-- This section is linked from [[Elias delta coding]] -->
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| Gamma coding does not code zero or negative integers.
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| One way of handling zero is to add 1 before coding and then subtract 1 after decoding.
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| Another way is to prefix each nonzero code with a 1 and then code zero as a single 0.
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| 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.
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| [[Exponential-Golomb coding]] generalizes the gamma code to integers with a "flatter" power-law distribution, just as [[Golomb coding]] generalizes the unary code.
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| 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.
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| ==References==
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| {{Reflist}}
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| * {{cite book | first = Khalid | last = Sayood | title = Lossless Compression Handbook | publisher = Elsevier | year = 2003 | chapter = Levenstein and Elias Gamma Codes | isbn = 978-0-12-620861-0}}
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| ==See also==
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| *[[Elias delta coding]]
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| *[[Elias omega coding]]
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| {{Compression Methods}}
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| {{DEFAULTSORT:Elias Gamma Coding}}
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| [[Category:Numeral systems]]
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| [[Category:Lossless compression algorithms]]
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| [[Category:Articles with example C code]]
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