Unary coding

Unary coding is an entropy encoding that represents a natural number, n, with n − 1 ones followed by a zero. For example 5 is represented as 11110. Some representations use n ones followed by a zero. The ones and zeros are interchangeable without loss of generality.

Unary coding is easily shown to be an optimally efficient encoding for the following discrete probability distribution

${\displaystyle \operatorname {P} (n)=2^{-n}\,}$

for ${\displaystyle n=1,2,3,...}$. It is in fact optimal for any geometric distribution

${\displaystyle \operatorname {P} (n)=(k-1)k^{-n}\,}$

for which k ≥ φ = 1.61803398879…, the golden ratio, or, more generally, for any distribution for which

${\displaystyle \operatorname {P} (n)\geq \operatorname {P} (n+1)+\operatorname {P} (n+2)\,}$

for ${\displaystyle n=1,2,3,...}$.

A modified unary encoding is used in UTF-8.