Unary coding

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 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.

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

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

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.

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 codes of 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.

Following is an example of uniquely decodable unary codes that is not a prefix code and is not instantaneously decodable ()

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 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.

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.