On Tilings of Asymmetric Limited-Magnitude Balls
Abstract
We study whether an asymmetric limited-magnitude ball may tile Zn. 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.
Turn this paper into a lesson
ArcXiv compiles a structured reading guide from this paper's metadata: plain-English importance, contributions, prerequisite concepts, which sections to read first, flashcards, and a quiz. Grounded in the abstract, never invented.