what is a generalized binary system? why 11 dimensions?


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merle van osdol
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Message 7979 - Posted 16 Feb 2010 22:50:03 UTC

    The title is my question. Can the answer be stated for one who has only gone thru calculus and a smattering of linear algebra? Your response would be greatly appreciated.

    kittka
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    Message 7980 - Posted 26 Feb 2010 16:47:44 UTC - in response to Message 7979.

      "The focus of the project is the investigation of generalized number systems (GNS). The usual decimal system can be generalized in several ways, GNS 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 project is to provide a complete list of GNS in a few special cases.
      Although there is 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 (Monatshefte article).
      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. The more special cases are covered, the better predictions we can make about the number and structure of systems with sizes currently unreachable by computational analysis."

      Thanks for the answer to Peter Burcsi and Attila Kovacs (ELTE)!

      merle van osdol
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      Message 7982 - Posted 5 Mar 2010 8:10:15 UTC

        Last modified: 5 Mar 2010 8:16:53 UTC

        Thanks for your explanation. I'll need some time to try and digest what you have given me here. Can you give me some examples of what you mean by all of this?

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