  This investigation studies the ergodic properties of reversible linear
cellular automata over $\mathbb{Z}_m$ for $m \in \mathbb{N}$. We show that a
reversible linear cellular automaton is either a Bernoulli automorphism or
non-ergodic. This gives an affirmative answer to an open problem proposed in
[Pivato, Ergodc theory of cellular automata, Encyclopedia of Complexity and
Systems Science, 2009, pp.~2980-3015] for the case of reversible linear
cellular automata.
