Unary coding

Short description: Entropy encoding

Unary coding,[nb 1] 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[clarification needed] encoding for the following discrete probability distribution

P⁡(n)=2−n

for n=1,2,3,....

In symbol-by-symbol coding, it is optimal for any geometric distribution

P⁡(n)=(k−1)k−n

for which k ≥ φ = 1.61803398879..., the golden ratio, or, more generally, for any discrete distribution for which

P⁡(n)≥P⁡(n+1)+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.

## Contents

* 1 Unary code in use today
* 2 Unary coding in biological networks
* 3 Standard run-length unary codes
* 4 Uniquely decodable non-prefix unary codes
* 5 Symmetric unary codes
* 6 Canonical unary codes
* 7 Generalized unary coding
* 8 See also
* 9 Notes
* 10 References

## 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.[2][3] 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' ( 2n−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'.[clarification needed]

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 | 000000000  | 111111111

Canonical codes can require less processing time to decode[clarification needed] 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.[4] 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.

## See also

* Unary numeral system

## 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. ↑ "University of Alberta Dictionary of Cognitive Science: Thermometer Code". http://www.bcp.psych.ualberta.ca/~mike/Pearl_Street/Dictionary/contents/T/thermcode.html.
2. ↑ "Neural network models of birdsong production, learning, and coding". New Encyclopedia of Neuroscience. Elsevier. 2007.
3. ↑ "Motor pathway convergence predicts syllable repertoire size in oscine birds". Proc. Natl. Acad. Sci. USA 108 (39): 16440–16445. 2011. doi:10.1073/pnas.1102077108. PMID 21918109. Bibcode: 2011PNAS..10816440M.
4. ↑ "Generalized unary coding". Circuits, Systems and Signal Processing 35 (4): 1419–1426. 2015. doi:10.1007/s00034-015-0120-7.

* v
* t
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Data compression methods

Lossless|

| Entropy type|

* Arithmetic
* Asymmetric numeral systems
* Golomb
* Huffman
* Adaptive
* Canonical
* Modified
* Range
* Shannon
* Shannon–Fano
* Shannon–Fano–Elias
* Tunstall
* Unary
* Universal
* Exp-Golomb
* Fibonacci
* Gamma
* Levenshtein

|

Dictionary type|

* Byte pair encoding
* Lempel–Ziv
* Brotli
* DEFLATE
* LZ4
* LZFSE
* LZJB
* LZMA
* LZO
* LZRW
* LZS
* LZSS
* LZW
* LZWL
* LZX
* Snappy
* Zstandard

Other types|

* BWT
* CTW
* Delta
* DMC
* DPCM
* LDCT
* MTF
* PAQ
* PPM
* RLE

Lossy|

| Transform type|

* Discrete cosine transform
* DCT
* MDCT
* DST
* FFT
* Wavelet
* Daubechies
* DWT
* SPIHT

|

Predictive type|

* DPCM
* ADPCM
* LPC
* ACELP
* CELP
* LAR
* LSP
* WLPC
* Motion
* Compensation
* Estimation
* Vector
* Psychoacoustic

Audio|

| Concepts|

* Bit rate
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* VBR
* Companding
* Convolution
* Dynamic range
* Latency
* Nyquist–Shannon theorem
* Sampling
* Sound quality
* Speech coding
* Sub-band coding

|

Codec parts|

* A-law
* μ-law
* DPCM
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* FT
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* LPC
* ACELP
* CELP
* LAR
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* WLPC
* MDCT
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Image|

| Concepts|

* Chroma subsampling
* Coding tree unit
* Color space
* Compression artifact
* Image resolution
* Macroblock
* Pixel
* PSNR
* Quantization
* Standard test image

|

Methods|

* Chain code
* DCT
* DEFLATE
* Fractal
* KLT
* LP
* RLE
* Wavelet
* Daubechies
* DWT
* EZW
* SPIHT

Video|

| Concepts|

* Bit rate
* ABR
* CBR
* VBR
* Display resolution
* Frame
* Frame rate
* Frame types
* Interlace
* Video characteristics
* Video quality

|

Codec parts|

* DCT
* DPCM
* Deblocking filter
* Lapped transform
* Motion
* Compensation
* Estimation
* Vector
* Wavelet
* Daubechies
* DWT

Theory|

* Entropy
* Information theory
* Timeline
* Kolmogorov complexity
* Quantization
* Rate–distortion
* Redundancy

* Compression formats
* Compression software (codecs)

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