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 kk, that is, with light bulbs in an n×nn×n formation, having kk different intensities cyclically switching from 00 (off) to (k−1)(k−1) (highest intensity) and then back to 00 (off). Under permutation switching, that is, adding a permutation matrix modulo kk, given a particular initial pattern, we investigate both the smallest number Rn,kRn,k of on-lights (the covering radius of the code generated) and the smallest total intensity In,kIn,k that can be attained. We obtain an explicit formula for In,kIn,k when nn is a multiple of kk. We also determine Rn,kRn,k when kk equals 2 and 3. In general, we obtain some bounds for Rn,kRn,k and In,kIn,k.