New arcs in projective Hjelmslev planes over Galois rings
Michael Kiermaier ; Axel Kohnert ;
in: Proceedings Optimal Codes and Related Topics 2007 Smart-Link: http://algorithm.uni-bayreuth.de/de/download/phg07.pdf
It is known that some good linear codes over a finite ring (R-linear codes) arise from interesting point constellations in certain projective geometries. For example, the expurgated Nordstrom-Robinson code, a nonlinear binary [14, 6, 6]-code which has higher minimum distance than any linear binary [14, 6]-code, can be constructed from a maximal 2-arc in the projective Hjelmslev plane over Z4. We report on a computer search for maximal arcs in projective Hjelmslev planes over proper Galois rings of order <= 27. The used method is to prescribe a group of automorphisms which shrinks the problem to a computationally feasible size. The resulting system of Diophantine linear equations is solved by lattice point enumeration. We improve many of the known lower bounds on the size of maximal arcs. Furthermore, the Gray image of one of the constructed arcs yields a quaternary [504, 6, 376]-code. This code has higher minimal distance than any known F4-linear [504, 6]-code.