On the non-existence of lattice tilings by quasi-crosses

Moshe Schwartz

doi:10.1109/ITA.2013.6502976

On the non-existence of lattice tilings by quasi-crosses

Moshe Schwartz

Department of Electrical & Computer Engineering

Research output: Contribution to conference › Paper › peer-review

2 Scopus citations

Abstract

We study necessary conditions for the existence of lattice tilings of ℝⁿ by quasi-crosses. We prove general non-existence results using a variety of number-theoretic tools. We then apply these results to the two smallest unclassified shapes, the (3, 1, n)-quasi-cross and the (3, 2, n)-quasi-cross. We show that for dimensions n ≤ 250, apart from the known constructions, there are no lattice tilings of ℝⁿ by (3, 1, n)-quasi-crosses except for ten remaining unresolved cases, and no lattice tilings of ℝⁿ by (3, 2, n)-quasi-crosses except for eleven remaining unresolved cases.

Original language	English
Pages	372-373
Number of pages	2
DOIs	https://doi.org/10.1109/ITA.2013.6502976
State	Published - 16 May 2013
Event	2013 Information Theory and Applications Workshop, ITA 2013 - San Diego, CA, United States Duration: 10 Feb 2013 → 15 Feb 2013

Conference

Conference	2013 Information Theory and Applications Workshop, ITA 2013
Country/Territory	United States
City	San Diego, CA
Period	10/02/13 → 15/02/13

ASJC Scopus subject areas

Computer Science Applications
Information Systems

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10.1109/ITA.2013.6502976

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title = "On the non-existence of lattice tilings by quasi-crosses",

abstract = "We study necessary conditions for the existence of lattice tilings of ℝn by quasi-crosses. We prove general non-existence results using a variety of number-theoretic tools. We then apply these results to the two smallest unclassified shapes, the (3, 1, n)-quasi-cross and the (3, 2, n)-quasi-cross. We show that for dimensions n ≤ 250, apart from the known constructions, there are no lattice tilings of ℝn by (3, 1, n)-quasi-crosses except for ten remaining unresolved cases, and no lattice tilings of ℝn by (3, 2, n)-quasi-crosses except for eleven remaining unresolved cases.",

author = "Moshe Schwartz",

note = "Funding Information: This work was supported in part by ISF grant 134/10 . ; 2013 Information Theory and Applications Workshop, ITA 2013 ; Conference date: 10-02-2013 Through 15-02-2013",

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AU - Schwartz, Moshe

N1 - Funding Information: This work was supported in part by ISF grant 134/10 .

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Y1 - 2013/5/16

N2 - We study necessary conditions for the existence of lattice tilings of ℝn by quasi-crosses. We prove general non-existence results using a variety of number-theoretic tools. We then apply these results to the two smallest unclassified shapes, the (3, 1, n)-quasi-cross and the (3, 2, n)-quasi-cross. We show that for dimensions n ≤ 250, apart from the known constructions, there are no lattice tilings of ℝn by (3, 1, n)-quasi-crosses except for ten remaining unresolved cases, and no lattice tilings of ℝn by (3, 2, n)-quasi-crosses except for eleven remaining unresolved cases.

AB - We study necessary conditions for the existence of lattice tilings of ℝn by quasi-crosses. We prove general non-existence results using a variety of number-theoretic tools. We then apply these results to the two smallest unclassified shapes, the (3, 1, n)-quasi-cross and the (3, 2, n)-quasi-cross. We show that for dimensions n ≤ 250, apart from the known constructions, there are no lattice tilings of ℝn by (3, 1, n)-quasi-crosses except for ten remaining unresolved cases, and no lattice tilings of ℝn by (3, 2, n)-quasi-crosses except for eleven remaining unresolved cases.

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DO - 10.1109/ITA.2013.6502976

M3 - Paper

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T2 - 2013 Information Theory and Applications Workshop, ITA 2013

Y2 - 10 February 2013 through 15 February 2013

ER -

On the non-existence of lattice tilings by quasi-crosses

Abstract

Conference

ASJC Scopus subject areas

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