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 − 1 zeros followed by a one. The ones and zeros are interchangeable without loss of generality.

n	coding
1	1
2	01
3	001
4	0001
5	00001
6	000001
7	0000001
8	00000001
9	000000001
10	0000000001

Unary coding is an optimally efficient encoding for the following discrete probability distribution

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

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

In symbol-by-symbol coding, it is 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 discrete distribution for which

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

for ${\displaystyle n=1,2,3,...}$. Although it is the optimal symbol-by-symbol coding for such probability distributions, its optimality can, like that of Huffman coding, be over-stated. Arithmetic coding has better compression capability for the last two distributions mentioned above because it does not consider input symbols independently, but rather implicitly groups the inputs.

A modified unary encoding is used in UTF-8. Unary codes are also used in split-index schemes like the Golomb Rice code. Unary coding is prefix-free, and can be uniquely decoded.

References

• Khalid Sayood, Data Compression, 3rd ed, Morgan Kaufmann.
• Professor K.R Rao, EE5359:Principles of Digital Video Coding.