  If A=Z/2, then A^Z is a compact abelian group. A `linear cellular automaton'
is a shift-commuting endomorphism F of A^Z. If P is a probability measure on
A^Z, then F `asymptotically randomizes' P if F^j P converges to the Haar
measure as j-->oo, for j in a subset of Cesaro density one. Via
counterexamples, we show that nonzero entropy of P is neither necessary nor
sufficient for asymptotic randomization.
