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
for .
In symbol-by-symbol coding, it is optimal for any geometric distribution
for which k ≥ φ = 1.61803398879…, the golden ratio, or, more generally, for any discrete distribution for which
for . 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.