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Construction of Two-Weight Codes
(2005)
- This is a talk given at the conference: Algebra and Computation 2005 in Tokyo. We describe a method for the construction of two-weight codes. This also allows to realize certain strongly regular graphs or equivalently certain point sets in the a finite projective geometry. We use the method of prescibed automorphisms, which allows us to reduce the problem to a size where we can use powerful Diophantine equation solvers provided by Alfred Wassermann.
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A note on Erdös-Diophantine graphs and Diophantine carpets
(2005)
- A Diophantine figure is a set of points on the integer grid $\mathbb{Z}^{2}$ where all mutual Euclidean distances are integers. We also speak of Diophantine graphs. The vertices are points in $\mathbb{Z}^{2}$ (the coordinates)and the edges are labeled with the distance between the two adjacent vertices, which is integral. In this language a Diophantine figure is a complete Diophantine graph. Two Diophantine graphs are equivalent if they only differ by translation or rotation of vertices. Due to a famous theorem of Erdös and Anning there are complete Diophantine graphs which are not contained in larger ones. We call them Erdös-Diophantine graphs. A special class of Diophantine graphs are Diophantine carpets. These are planar triangulations of a subset of the integer grid. We give an effective construction for Erdös-Diophantine graphs and characterize the chromatic number of Diophantine carpets.
