Unary coding

 ( Coding Theory ) 

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. 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:

1	0	1
01	1	2
001	2	3
0001	3	4
00001	4	5
000001	5	6
0000001	6	7
00000001	7	8
000000001	8	9
0000000001	9	10

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

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

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

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

$\backslash operatorname\{P\}(n)\; =\; (k-1)k^\{-n\}\backslash ,$

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

$\backslash operatorname\{P\}(n)\; \backslash ge\; \backslash operatorname\{P\}(n+1)\; +\; \backslash operatorname\{P\}(n+2)\backslash ,$

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.

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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.

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Unary coding in biological networks Unary coding is used in the 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.

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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.

0
00
000
0000
00000
000000
0000000
00000000
000000000
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.

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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)

1	0
10	01
100	011
1000	0111
10000	01111
100000	011111
1000000	0111111
10000000	01111111
100000000	011111111
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.

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Symmetric unary codes

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

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
...

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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. The largest n numerical '0' or '-1' ( $\backslash operatorname2^\{n\}\; -\; 1\backslash ,$) 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'.

1	0
01	10
001	110
0001	1110
00001	11110
000001	111110
0000001	1111110
00000001	11111110
000000001	111111110
000000000	111111111

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.

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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.

0000000
0000111
0001110
0011100
0111000
1110000
0010111
0101110
1011100
0111001
1110010
0100111
1001110
0011101
0111010
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.

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See also

• Unary numeral system

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Notes

Categories: Coding Theory, Entropy Coding, Data Compression

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