# 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

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

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.[1][2] The nucleus in the brain of the songbirds that plays a part in both the learning and the production of bird song is the HVC (high vocal center). The use of unary n 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.[3]

## References

1. ^ Fiete, I.R. and H.S. Seung, Neural network models of birdsong production, learning, and coding. New Encyclopediaof Neuroscience. Eds. L. Squire, T. Albright, F. Bloom, F. Gage, and N. Spitzer. Elsevier, 2007.
2. ^ Moore J.M. et al., Motor pathway convergence predicts syllable repertoire size in oscine birds. Proc. Nat. Acad. Sc. USA 108: 16440-16445, 2011.
3. ^ Potluri, P., Error correction capacity of unary coding. 2014.http://arxiv.org/abs/1411.7406