Unary coding

Unary coding, or the unary numeral system and also 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 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.

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
1	1
2	10
3	100
4	1000
5	10000
6	100000
7	1000000
8	10000000
9	100000000
10	1000000000
11	10000000000
12	100000000000
13	1000000000000
14	10000000000000
15	100000000000000
....	10.......................00

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 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 to be pre-specified because this range determines the number of bits that are needed.

See also

• Unary numeral system