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. Unary coding is both a Prefix-free code and a Self-synchronizing code.

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.

Unary code in use today

Examples of unary code uses include:

• In Golomb Rice code, unary encoding is used to encode the quotient part of the Golomb code word.
• In UTF-8, unary encoding is used in the leading byte of a multi-byte sequence to indicates the number of bytes in the sequence, so that the length of the sequence can be determined without examining the continuation bytes.
• Instantaneously trained neural networks use unary coding for efficient data representation.

Unary coding in biological networks

New research has shown that unary coding is used in the neural circuits responsible for birdsong production. This use in biological networks is presumably due to the inherent simplicity of the coding. Another contributing factor could be the fact that unary coding provides a certain degree of error correction.

See also

• Unary numeral system