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On the minimum diameter of plane integral point sets
(2007)

Sascha Kurz
Alfred Wassermann
 Since ancient times mathematicians consider geometrical objects with integral side lengths. We consider plane integral point sets P, which are sets of n points in the plane with pairwise integral distances where not all the points are collinear. The largest occurring distance is called its diameter. Naturally the question about the minimum possible diameter d(2,n) of a plane integral point set consisting of n points arises. We give some new exact values and describe stateoftheart algorithms to obtain them. It turns out that plane integral point sets with minimum diameter consist very likely of subsets with many collinear points. For this special kind of point sets we prove a lower bound for d(2,n) achieving the known upper bound n^{c_2loglog n} up to a constant in the exponent.

Double and bordered alphacirculant selfdual codes over finite commutative chain rings
(2008)

Michael Kiermaier
Alfred Wassermann
 In this paper we investigate codes over finite commutative rings R, whose generator matrices are built from alphacirculant matrices. For a nontrivial ideal I < R we give a method to lift such codes over R/I to codes over R, such that some isomorphic copies are avoided. For the case where I is the minimal ideal of a finite chain ring we refine this lifting method: We impose the additional restriction that lifting preserves selfduality. It will be shown that this can be achieved by solving a linear system of equations over a finite field. Finally we apply this technique to Z_4linear double negacirculant and bordered circulant selfdual codes. We determine the best minimum Lee distance of these codes up to length 64.