On Tilings of Asymmetric Limited-Magnitude Balls

05/30/2020

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by   Hengjia Wei, et al.

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We study whether an asymmetric limited-magnitude ball may tile ℤ^n. 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.