A Gale-Berlekamp permutation-switching problem

In the spirit of the light switching game of Gale and Berlekamp, we define a light switching game based on permutations. We consider the game over the integers modulo k, that is, with light bulbs in an n × n formation, having k different intensities cyclically switching from 0 (off) to (k − 1) (highest intensity) and then back to 0 (off). Under permutation switching, that is, adding a permutationmatrix modulo k, given a particular initial pattern, we investigate both the smallest number Rn,k of on-lights (the covering radius of the code generated) and the smallest total intensityIn,k that can be attained. We obtain an explicit formula for In,k when n is a multiple of k. We also determine Rn,k when k equals 2 and 3. In general, we obtain some bounds for Rn,k and In,k. © 2014 Elsevier Ltd. All rights reserved.