BinSYS Project

Aims

The aim of the project is to find all the generalized binary number systems up to dimension 11. Below we give a short description of the number system concept and mention a few possible applications.

Introduction

Let n be an integer greater than one. When we speak of number systems in the original sense, we use the fact that each natural number z can be written uniquely in the finite form

_{}.

We say that n is the base
of the number
system, the _{} are
called the digits. If n = 2 then we speak
of a binary number system. These systems are
too poor to represent negative
numbers so we need a sign.
If we allow
the base to be a negative integer, a representation of all integers may
become possible. Such, for example
if we use
the base -2, each integer has a form

_{}

This can be generalized for the algebraic integers of a finite extension of the rational number field. A simple example: all the Gaussian integers (complex numbers of the form x+yi, where x,y are integers) can be written uiquely in the base (-1+i) as follows

_{}.

Using linear algebra we can define number systems in an even more general way. The base is now a matrix and the digits are vectors. We can reformulate the previous example. Each two-dimensional integer vector has a representation as a finite sum

_{},

where

_{} and _{}.

We speak of a binary system
if the determinant
of M is ±2. In this case there
are only two digits, one
of them being the origin. This
means that if we have
a number system then every integer vector can be represented
as a finite series of 0s and
1s.

Not every matrix M can be a number system base. Until now no characterisation of ”good” matrices have been given. There are sufficient conditions and there are necessary ones but the gap between them is too large. There is no known efficient method of dealing with matrices that fulfil necessary conditions but fail sufficient conditions. One thing to note is that if we fix the determinant and the dimension then roughly speaking there are only a finite number of possible matrices.

Expected results

The program aims at finding many generalized binary number systems. An extensive search is performed in the finite set of matrices of given size fulfilling some necessary conditions. The difficulty is that the size of this finite set is an exponantial function of the dimension. It now seems possible to attack the case of 11 ´ 11 matrices. To check further necessary conditions the program performs a lot of floating-point calculation. Thus, a lot of CPU time is needed. Luckily, parallelization is possible and we can benefit of running on several machines.

The program outputs a list of matrices (being more precise characteristic polynomials) that are already likely to be number system bases. This list is processed by another program (which does not need so much CPU). The final result is then a (complete) list of binary number systems in a fixed dimension.

Thereafter we perform information theoretical analysis. The number systems provide a binary representation of integer vectors. Using coordinates we have another (more standard) representation. The two representations usually differ in length. Besides, vectors close to each other in the space can have binary representations that look very different. These observations suggest that one could apply number systems in data compression, coding or cryptography.

Number systems are interesting from a geometrical point of view, too. If we allow negative powers of M to appear in the binary representation we get a possibly infinite representation of real vectors (we could say that we use a radix point). The boundary of the set of vectors with binary representation containing only negative powers of M (the set H of numbers with zero integer part) has mostly fractional dimension (it is a „fractal”). The output of the program can be used to analyze these sets. This means topologocal analysis, e.g. calculation of the dimension, connectedness etc. If we use the matrix M above, we get the following set.

Finally, knowing all matrices
up to a given
dimension could help us to
a deeper understanding of the mathematics
of generalized number systems.