A Gale-Berlekamp permutation-switching problem in higher dimensions

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Let an n× n array (aij) of lights be given, each either on (when aij=1) or off (when aij=-1). For each row and each column there is a switch so that if the switch is pulled (xᵢ=-1 for row i and yⱼ=-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.