Unary coding

Unary coding, sometimes called thermometer code, is an entropy encoding that represents a natural number, n, with n ones followed by a zero (if natural number is understood as non-negative integer) or with n − 1 ones followed by a zero (if natural number is understood as strictly positive integer). For example 5 is represented as 111110 or 11110. Some representations use n or n − 1 zeros followed by a one. The ones and zeros are interchangeable without loss of generality.

n (non-negative)	n (strictly positive)	Unary code	Alternative
0	1	0	1
1	2	10	01
2	3	110	001
3	4	1110	0001
4	5	11110	00001
5	6	111110	000001
6	7	1111110	0000001
7	8	11111110	00000001
8	9	111111110	000000001
9	10	1111111110	0000000001

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

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

for $n=1,2,3,...$.

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

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

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

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

for $n=1,2,3,...$. 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.

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.

See also

• Unary numeral system

References

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