A Gale-Berlekamp permutation-switching problem in higher dimensions

TL;DR

This paper generalizes the Gale-Berlekamp permutation-switching problem to higher dimensions, providing exact parameters for the unbalancing lights problem in these complex settings.

Contribution

It introduces a higher-dimensional extension of the unbalancing lights problem and derives exact parameters for this generalized scenario.

Findings

Exact parameters for higher-dimensional unbalancing lights problem

Extension of Gale-Berlekamp switching game to multiple dimensions

Theoretical framework for analyzing multi-dimensional switch configurations

Abstract

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.