English
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
Doi-Nummer:
Abstract:
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.
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