Unary coding,[nb 1] or the unary numeral system and also sometimes called thermometer code, is an entropy encoding that represents a natural number, n, with a code of length n + 1 ( or n ), usually 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
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.
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