For n=m=p prime, the only Ducci periods are 1, the order of 2 modulo p (for constant tuples), and p times that order; for n=3 and m odd prime, all non-exceptional tuples realize the maximum period.
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Values of Ducci Periods for Sequences on $\mathbb{Z}_m^n$
For n=m=p prime, the only Ducci periods are 1, the order of 2 modulo p (for constant tuples), and p times that order; for n=3 and m odd prime, all non-exceptional tuples realize the maximum period.