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

$\operatorname {P} (n)=2^{-n}\,$ 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

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

Unary coding is used in the neural circuits responsible for birdsong production. 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 command signals for different notes in the birdsong emanate from different points in the HVC. This coding works as space coding which is an efficient strategy for biological circuits due to its inherent simplicity and robustness.

## Standard run-length unary codes

All binary data is defined by the ability to represent unary numbers in alternating run-lengths of 1s and 0s. This conforms to the standard definition of unary ie N digits of the same number 1 or 0. All run-lengths by definition have at least one digit and thus represent strictly positive integers.

n RL code Next code
1 1 0
2 11 00
3 111 000
4 1111 0000
5 11111 00000
6 111111 000000
7 1111111 0000000
8 11111111 00000000
9 111111111 000000000
10 1111111111 0000000000
...

These codes are guaranteed to end validly on any length of data ( when reading arbitrary data ) and in the ( separate ) write cycle allow for the use and transmission of an extra bit ( the one used for the first bit ) while maintaining overall and per-integer unary code lengths of exactly N.

## Uniquely decodable non-prefix unary codes

Following is an example of uniquely decodable unary codes that is not a prefix code and is not instantaneously decodable (need look-ahead to decode)

n Unary code Alternative
1 1 0
2 10 01
3 100 011
4 1000 0111
5 10000 01111
6 100000 011111
7 1000000 0111111
8 10000000 01111111
9 100000000 011111111
10 1000000000 0111111111
...

These codes also ( when writing unsigned integers ) allow for the use and transmission of an extra bit ( the one used for the first bit ). Thus they are able to transmit 'm' integers * N unary bits and 1 additional bit of information within m*N bits of data.

## Symmetric unary codes

Following set of unary codes are symmetric and can be read in any direction. It is also instantaneously decodable in either direction.

n (strictly positive) Unary code Alternative n (non-negative)
1 1 0 0
2 00 11 1
3 010 101 2
4 0110 1001 3
5 01110 10001 4
6 011110 100001 5
7 0111110 1000001 6
8 01111110 10000001 7
9 011111110 100000001 8
10 0111111110 1000000001 9
...

## Canonical unary codes

For unary values where the maximum is known, one can use canonical unary codes that are of a somewhat numerical nature and different from character based codes. It involves starting with numerical '0' or '-1' ( $\operatorname {2} ^{n}-1\,$ ) and the maximum number of digits then for each step reducing the number of digits by one and increasing/decreasing the result by numerical '1'.

n Unary code Alternative
1 1 0
2 01 10
3 001 110
4 0001 1110
5 00001 11110
6 000001 111110
7 0000001 1111110
8 00000001 11111110
9 000000001 111111110
10 0000000000 1111111111

Canonical codes can require less processing time to decode when they are processed as numbers not a string. If the number of codes required per symbol length is different to 1, ie there are more non-unary codes of some length required, those would be achieved by increasing/decreasing the values numerically without reducing the length in that case.

## Generalized unary coding

A generalized version of unary coding was presented by Subhash Kak to represent numbers much more efficiently than standard unary coding. Here's an example of generalized unary coding for integers from 0 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 to be pre-specified because this range determines the number of bits that are needed.