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 nonnegative 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 Prefixfree code and a Selfsynchronizing code.
n (nonnegative)

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 symbolbysymbol coding, it is optimal for any geometric distribution

\operatorname{P}(n) = (k1)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 symbolbysymbol 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.
Contents

Unary code in use today 1

Unary coding in biological networks 2

Generalized unary coding 3

See also 4

References 5
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 UTF8, unary encoding is used in the leading byte of a multibyte sequence to indicate 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). This coding works as space coding which is an efficient strategy for biological circuits due to its inherent simplicity and robustness.
Generalized unary coding
A generalized version of unary coding is able to represent numbers much more efficiently than standard unary coding.^{[3]} Here's an example of generalized unary coding for integers from 1 through 15 that requires only 7 bits (where three bits are arbitrarily chosen in place of a single one in standard unary to show the number). Note that the representation is cyclic where one uses markers to represent higher integers in higher cycles.
n

Unary code

Generalized unary

0

0

0000000

1

10

0000111

2

110

0001110

3

1110

0011100

4

11110

0111000

5

111110

1110000

6

1111110

0010111

7

11111110

0101110

8

111111110

1011100

9

1111111110

0111001

10

11111111110

1110010

11

111111111110

0100111

12

1111111111110

1001110

13

11111111111110

0011101

14

111111111111110

0111010

15

1111111111111110

1110100

Generalized unary coding requires that the range of numbers to be represented be prespecified because this range determines the number of bits that are needed.
See also
References

^ 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.

^ Moore J.M. et al., Motor pathway convergence predicts syllable repertoire size in oscine birds. Proc. Nat. Acad. Sc. USA 108: 1644016445, 2011.

^ Kak, S., Generalized unary coding. Circuits, Systems and Signal Processing. 2015. http://link.springer.com/article/10.1007/s0003401501207#page1
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