2 research outputs found
On Tilings of Asymmetric Limited-Magnitude Balls
We study whether an asymmetric limited-magnitude ball may tile
. This ball generalizes previously studied shapes: crosses,
semi-crosses, and quasi-crosses. Such tilings act as perfect error-correcting
codes in a channel which changes a transmitted integer vector in a bounded
number of entries by limited-magnitude errors.
A construction of lattice tilings based on perfect codes in the Hamming
metric is given. Several non-existence results are proved, both for general
tilings, and lattice tilings. A complete classification of lattice tilings for
two certain cases is proved
On Lattice Packings and Coverings of Asymmetric Limited-Magnitude Balls
We construct integer error-correcting codes and covering codes for the
limited-magnitude error channel with more than one error. The codes are
lattices that pack or cover the space with the appropriate error ball. Some of
the constructions attain an asymptotic packing/covering density that is
constant. The results are obtained via various methods, including the use of
codes in the Hamming metric, modular -sequences, -fold Sidon sets, and
sets avoiding arithmetic progression