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 codeAlternative
0101
121001
23110001
3411100001
451111000001
56111110000001
6711111100000001
781111111000000001
89111111110000000001
91011111111100000000001

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

for .

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 . 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.[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). 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.

nRL codeNext code
110
21100
3111000
411110000
51111100000
6111111000000
711111110000000
81111111100000000
9111111111000000000
1011111111110000000000
...

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)

nUnary code Alternative
11 0
210 01
3100 011
41000 0111
510000 01111
6100000 011111
71000000 0111111
810000000 01111111
9100000000 011111111
101000000000 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)
11 0 0
200 11 1
3010 101 2
40110 1001 3
501110 10001 4
6011110 100001 5
70111110 1000001 6
801111110 10000001 7
9011111110 100000001 8
100111111110 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' ( ) 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'.

nUnary code Alternative
11 0
201 10
3001 110
40001 1110
500001 11110
6000001 111110
70000001 1111110
800000001 11111110
9000000001 111111110
100000000000 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, 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.[3] 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.

nUnary codeGeneralized unary
000000000
1100000111
21100001110
311100011100
4111100111000
51111101110000
611111100010111
7111111100101110
81111111101011100
911111111100111001
10111111111101110010
111111111111100100111
1211111111111101001110
13111111111111100011101
141111111111111100111010
1511111111111111101110100

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

Notes

  1. The equivalent to the term "unary coding" in German scientific literature is "BCD-Zählcode", which would translate into "binary-coded decimal counting code". This must not be confused with the similar German term "BCD-Code" translating to BCD code in English.

References

  1. Fiete, I. R.; Seung, H. S. (2007). "Neural network models of birdsong production, learning, and coding". In Squire, L.; Albright, T.; Bloom, F.; Gage, F.; Spitzer, N. (eds.). New Encyclopedia of Neuroscience. Elsevier.
  2. Moore, J. M.; et al. (2011). "Motor pathway convergence predicts syllable repertoire size in oscine birds". Proc. Natl. Acad. Sci. USA. 108 (39): 16440–16445. Bibcode:2011PNAS..10816440M. doi:10.1073/pnas.1102077108. PMC 3182746. PMID 21918109.
  3. Kak, S. (2015). "Generalized unary coding". Circuits, Systems and Signal Processing. 35 (4): 1419–1426. doi:10.1007/s00034-015-0120-7. S2CID 27902257.
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