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 permutation matrix modulo k , given a particular initial pattern, we investigate both the smallest number R n , k of on-lights (the covering radius of the code generated) and the smallest total intensity I n , k that can be attained. We obtain an explicit formula for I n , k when n is a multiple of k . We also determine R n , k when k equals 2 and 3. In general, we obtain some bounds for R n , k and I n , k .