Data Compression Using Asymmetric Numeral Systems

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram CDF Player or other Wolfram Language products.

Requires a Wolfram Notebook System

Edit on desktop, mobile and cloud with any Wolfram Language product.

This Demonstration allows you to experiment with compression using a new method: asymmetric numeral systems.


Using a probability distribution , a sequence of symbols of a given length is generated and encoded using the parameters of the encoder.

The decoding table, starting from a given state, produces a symbol and new state, with blue bits added to get the new state in the range.

A summary is on the right: the optimal asymptotic minimum bit per symbol required (Shannon's entropy), how it should be asymptotically compressed with an ideal compressor but using no necessarily optimal (=) probability distribution, and how it compares with the generated example. Next you see the distribution of symbols in the table, the generated sequence, and finally the correlations in encoded sequences of bits.


Contributed by: Jarek Duda (March 2011)
Open content licensed under CC BY-NC-SA



Numeral systems are optimal for encoding sequences of symbols (digits) that have uniform distribution.

An asymmetric numeral system is a generalization constructed to be optimal for any given probability distribution of symbols (digits).

With some information stored in a natural number , to add some information stored in a digit that all has the same probability (), take to place this information in the right-most bits.

In the asymmetric case should hold asymptotically.

This cannot usually be done exactly, but for asymptotic behavior the symbols only need to be distributed uniformly. A random number generator can be used to choose a specific distribution. If this generator is initialized with a given key, this additionally encrypts the output.

To encode in this way, would grow to infinity. To prevent that, is restricted to some range and some bits are removed in blocks of "width" bits to a bit stream (compressed file).

Basic information about this coding can be found in:

More information about using it in cryptography:

Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.