I received a few information about the project status. There is a short description and the results we have already achieved.
The usual decimal system can be generalized in several ways, GNS (generalized number systems) being one of them. In these systems the decimal base 10 is replaced by a square matrix and the digits 0-9 are replaced by a suitable set of vectors, also called digits. In some cases, every vector in the space can be represented by a finite sum of powers of the base each multiplied by one of the digits, the same way as for example 256 is the sum of 100 times 2, 10 times 5 and 1 times 6. In these cases the vectors of the space can be mapped to finite sequences of digits and vice versa - suggesting potential applications in coding or cryptography. These applications explain why these systems are of interest for informatics.
Unfortunately, the exact conditions on the base and the vector that ensure that such a representation exists and is unique are currently unknown, making the problem also interesting for mathematicians. The aim of the application is to provide a complete list of GNS in a few special cases. Although there are an infinite number of systems, if we fix the size of the problem, that is the size and number of the vectors and impose an additional condition of using the so-called canonical digit set, then this number is finite and computer methods for traversing the search space exist. Obtaining the complete lists for the special cases will hopefully support the better mathematical understanding of the structure of GNS. Some partial results are already obtained in Burcsi, P., Kovács, A.: Exhaustive search methods for CNS polynomials, Monatshefte für Mathematik, 155, 421-430, 2008
Currently we know all canonical binary number systems up to dimension 13 (the former record was 8), and many ternary number systems. The research also led to the invention of the concept of semi-number-systems. |