Equal moments division of a set

Let N^*_q(m) be the minimal positive integer N, for which there exists a splitting of the set [0, N - 1] into q subsets, S ₀, S₁, ..., S_q-1, whose first m moments are equal. Similarly, let m^*_q(N) be the maximal positive integer m, such that there exists a splitting of [0, N - 1] into q subsets whose first m moments are equal. For q = 2, these functions were investigated by several authors, and the values of N^*₂(m) and m^*₃(N) have been found for m ≤ 8 and N ≤ 167, respectively. In this paper, we deal with the problem for any prime q. We demonstrate our methods by finding m^*₃(N) for any N < 90 and N^*₃ (m) for m ≤ 6.