A Gale-Berlekamp permutation-switching problem in higher dimensions

Let an $n\times n$ array $\left( a_{ij}\right) $ of lights be given, each either on (when $a_{ij}=1$) or off (when $a_{ij}=-1$). For each row and each column there is a switch so that if the switch is pulled ($x_{i}=-1$ for row $i$ and $y_{j}=-1$ for column $j$) all of the lights in that line are sw...

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Bibliographic Details
Main Authors Araujo, Gustavo, Pellegrino, Daniel Marinho
Format Journal Article
LanguageEnglish
Published 28.01.2018
Subjects
Mathematics - Combinatorics
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Summary:Let an $n\times n$ array $\left( a_{ij}\right) $ of lights be given, each either on (when $a_{ij}=1$) or off (when $a_{ij}=-1$). For each row and each column there is a switch so that if the switch is pulled ($x_{i}=-1$ for row $i$ and $y_{j}=-1$ for column $j$) all of the lights in that line are switched: on to off or off to on. The unbalancing lights problem (Gale-Berlekamp switching game) consists in maximizing the difference between the lights on and off. We obtain the exact parameters for a generalization of the unbalancing lights problem in higher dimensions.
DOI:10.48550/arxiv.1801.09194