Exact hyperplane covers for subsets of the hypercube

James Aaronson; Carla Groenland; Andrzej Grzesik; Tom Johnston; Bartłomiej Kielak

doi:10.1016/j.disc.2021.112490

Exact hyperplane covers for subsets of the hypercube

James Aaronson, Carla Groenland, Andrzej Grzesik, Tom Johnston, Bartłomiej Kielak

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

Abstract

Alon and Füredi (1993) showed that the number of hyperplanes required to cover {0,1}ⁿ∖{0} without covering 0 is n. We initiate the study of such exact hyperplane covers of the hypercube for other subsets of the hypercube. In particular, we provide exact solutions for covering {0,1}ⁿ while missing up to four points and give asymptotic bounds in the general case. Several interesting questions are left open.

Original language	English
Article number	112490
Pages (from-to)	1-7
Journal	Discrete Mathematics
Volume	344
Issue number	9
DOIs	https://doi.org/10.1016/j.disc.2021.112490
Publication status	Published - Sept 2021

Keywords

Exact covers
Hypercube
Hyperplanes
Intersection patterns

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10.1016/j.disc.2021.112490Licence: CC BY-NC-ND

1-s2.0-S0012365X2100203X-mainFinal published version, 259 KBLicence: CC BY-NC-ND

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abstract = "Alon and F{\"u}redi (1993) showed that the number of hyperplanes required to cover {0,1}n∖{0} without covering 0 is n. We initiate the study of such exact hyperplane covers of the hypercube for other subsets of the hypercube. In particular, we provide exact solutions for covering {0,1}n while missing up to four points and give asymptotic bounds in the general case. Several interesting questions are left open.",

keywords = "Exact covers, Hypercube, Hyperplanes, Intersection patterns",

author = "James Aaronson and Carla Groenland and Andrzej Grzesik and Tom Johnston and Bart{\l}omiej Kielak",

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AU - Grzesik, Andrzej

AU - Johnston, Tom

AU - Kielak, Bartłomiej

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N2 - Alon and Füredi (1993) showed that the number of hyperplanes required to cover {0,1}n∖{0} without covering 0 is n. We initiate the study of such exact hyperplane covers of the hypercube for other subsets of the hypercube. In particular, we provide exact solutions for covering {0,1}n while missing up to four points and give asymptotic bounds in the general case. Several interesting questions are left open.

AB - Alon and Füredi (1993) showed that the number of hyperplanes required to cover {0,1}n∖{0} without covering 0 is n. We initiate the study of such exact hyperplane covers of the hypercube for other subsets of the hypercube. In particular, we provide exact solutions for covering {0,1}n while missing up to four points and give asymptotic bounds in the general case. Several interesting questions are left open.

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KW - Hyperplanes

KW - Intersection patterns

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JO - Discrete Mathematics

JF - Discrete Mathematics

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Exact hyperplane covers for subsets of the hypercube

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Keywords

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