On Tilings of Asymmetric Limited-Magnitude Balls.

Hengjia Wei; Moshe Schwartz

doi:10.1016/j.ejc.2021.103450

On Tilings of Asymmetric Limited-Magnitude Balls.

Hengjia Wei, Moshe Schwartz

Department of Electrical & Computer Engineering

Ben-Gurion University of the Negev

Research output: Contribution to journal › Article › peer-review

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.

Original language	American English
Article number	103450
Pages (from-to)	1-21
Number of pages	21
Journal	European Journal of Combinatorics
Volume	100
DOIs	https://doi.org/10.1016/j.ejc.2021.103450
State	Published - 1 Feb 2022

All Science Journal Classification (ASJC) codes

Discrete Mathematics and Combinatorics

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10.1016/j.ejc.2021.103450

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title = "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.",

author = "Hengjia Wei and Moshe Schwartz",

note = "Funding Information: This work was supported in part by an Israel Science Foundation (ISF) grant 270/18 . Publisher Copyright: {\textcopyright} 2021 Elsevier Ltd",

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AB - 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.

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ER -

On Tilings of Asymmetric Limited-Magnitude Balls.

Abstract

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