Unary coding,^{[nb 1]} or theunary numeral system, is anentropy encoding that represents anatural number,n, withn ones followed by a zero (if the termnatural number is understood asnon-negative integer) or withn − 1 ones followed by a zero (if the termnatural number is understood asstrictly positive integer). A unary number's code length would thus ben + 1 with that first definition, orn with that second definition. Unary code when vertical behaves like mercury in athermometer that gets taller or shorter asn gets bigger or smaller, and so is sometimes calledthermometer code.^[1] An alternative representation usesn orn − 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 coding is anoptimally efficient^{[clarification needed]} encoding for the following discreteprobability distribution^{[citation needed]}

${\displaystyle \mathrm{P}(n)=2^{-n}}$

for${\displaystyle n=1,2,3,...}$.

In symbol-by-symbol coding, it is optimal for anygeometric distribution

${\displaystyle \mathrm{P}(n)=(k-1)k^{-n}}$

for whichk ≥ φ = 1.61803398879..., thegolden ratio, or, more generally, for any discrete distribution for which

${\displaystyle \mathrm{P}(n)\ge \mathrm{P}(n+1)+\mathrm{P}(n+2)}$

for${\displaystyle n=1,2,3,...}$. 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 aprefix-free code and aself-synchronizing code.

Examples of unary code uses include:

• InGolomb Rice code, unary encoding is used to encode the quotient part of the Golomb code word.
• InUTF-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 is used in theneural circuits responsible forbirdsong 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.

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 representstrictly positive integers.

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.

Following is an example ofuniquely decodable unary codes that is not aprefix code and is not instantaneously decodable (need look-ahead to decode)

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.

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

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 largestn numerical '0' or '-1' (${\displaystyle {\mathrm{2}}^n-1}$) 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]}

Canonical codes canrequire 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.

A generalized version of unary coding was presented bySubhash 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.

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.

• Unary numeral system

• Coding theory
• Entropy coding
• Data compression

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