Unary coding explained

Unary coding, or the unary numeral system, is an entropy encoding that represents a natural number, n, with n ones followed by a zero (if the term natural number is understood as non-negative integer) or with n - 1 ones followed by a zero (if the term natural number is understood as strictly positive integer). A unary number's code length would thus be n + 1 with that first definition, or n with that second definition. Unary code when vertical behaves like mercury in a thermometer that gets taller or shorter as n gets bigger or smaller, and so is sometimes called thermometer code.^[1] An alternative representation uses n or n - 1 zeros followed by a one, effectively swapping the ones and zeros, without loss of generality. For example, the first ten unary codes are:

Unary code	Alternative	n (non-negative)	n (strictly positive)
0	1	0	1
10	01	1	2
110	001	2	3
1110	0001	3	4
11110	00001	4	5
111110	000001	5	6
1111110	0000001	6	7
11111110	00000001	7	8
111111110	000000001	8	9
1111111110	0000000001	9	10

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)\ge\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 coding is both a prefix-free code and a self-synchronizing code.

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 i.e. 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

The following symmetric unary codes can be read and instantaneously decoded in either direction:

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

Canonical unary codes

See also: Canonical Huffman code. 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. The largest n numerical '0' or '-1' (

\operatorname2ⁿ-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	0000000001	1111111110

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, i.e. 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.

Notes and References

Web site: University of Alberta Dictionary of Cognitive Science: Thermometer Code . 2025-05-31 . www.bcp.psych.ualberta.ca.