Editing Unary coding (section)

{{Short description|Entropy encoding}}
'''Unary coding''',<ref group="nb" name="NB1"/> or the [[unary numeral system]], is an [[entropy encoding]] that represents a [[natural number]], ''n'', with ''n'' ones followed by a zero (if the term ''natural number'' is understood as ''non-negative integer'') or with ''n''&nbsp;&minus;&nbsp;1 ones followed by a zero (if the term ''natural number'' is understood as ''strictly positive integer''). A unary number's code length would thus be ''n''&nbsp;+&nbsp;1 with that first definition, or ''n'' with that second definition. Unary code when vertical behaves like mercury in a [[thermometer]] that gets taller or shorter as ''n'' gets bigger or smaller, and so is sometimes called '''thermometer code'''.<ref>{{Cite web |title=University of Alberta Dictionary of Cognitive Science: Thermometer Code |url=http://www.bcp.psych.ualberta.ca/~mike/Pearl_Street/Dictionary/contents/T/thermcode.html |access-date=2025-05-31 |website=www.bcp.psych.ualberta.ca}}</ref> An alternative representation uses ''n'' or ''n''&nbsp;&minus;&nbsp;1 zeros followed by a one, effectively swapping the ones and zeros, [[without loss of generality]]. For example, the first ten unary codes are:
{| class="wikitable"
! Unary code !! Alternative
!n (non-negative)
!n (strictly positive)
|-
| 0 || 1
|0
|1
|-
| 10 || 01
|1
|2
|-
| 110 || 001
|2
|3
|-
| 1110 || 0001
|3
|4
|-
| 11110 || 00001
|4
|5
|-
| 111110 || 000001
|5
|6
|-
| 1111110 || 0000001
|6
|7
|-
| 11111110 || 00000001
|7
|8
|-
| 111111110 || 000000001
|8
|9
|-
| 1111111110 || 0000000001
|9
|10
|}

Unary coding is an ''optimally efficient''{{Clarify|reason=This term needs some introduction.|date=May 2025}} encoding for the following discrete [[probability distribution]]{{Citation needed|date=May 2025|reason=Citation needed for this strong claim and to explain this term "optimally efficient".}}

:<math>\operatorname{P}(n) = 2^{-n}\,</math>

for <math>n=1,2,3,...</math>.

In symbol-by-symbol coding, it is optimal for any [[geometric distribution]]

:<math>\operatorname{P}(n) = (k-1)k^{-n}\,</math>

for which ''k'' &ge; &phi; = 1.61803398879..., the [[golden ratio]], or, more generally, for any discrete distribution for which

:<math>\operatorname{P}(n) \ge \operatorname{P}(n+1) + \operatorname{P}(n+2)\, </math>

for <math>n=1,2,3,...</math>.  Although it is the optimal symbol-by-symbol coding for such probability distributions, [[Golomb coding]] achieves better compression capability for the geometric distribution because it does not consider input symbols independently, but rather implicitly groups the inputs. For the same reason, [[arithmetic encoding]] performs better for general probability distributions, as in the last case above.

Unary coding is both a [[prefix-free code]] and a [[self-synchronizing code]].