Unary coding is an entropy encoding that represents a natural number, n, with n − 1 ones followed by a zero. For example 5 is represented as 11110. Some representations use n − 1 zeros followed by a one. The ones and zeros are interchangeable without loss of generality.

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

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

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 n = 1,2,3,.... Although it is the optimal symbol-by-symbol coding for such probability distributions, its optimality can, like that of Huffman coding, be over-stated. Arithmetic coding has better compression capability for the last two distributions mentioned above because it does not consider input symbols independently, but rather implicitly groups the inputs.

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.

References

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