  The properties of maximum Lempel-Ziv complexity strings are studied for the
binary case. A comparison between MLZs and random strings is carried out. The
length profile of both type of sequences show different distribution functions.
The non-stationary character of the MLZs are discussed. The issue of
sensitiveness to noise is also addressed. An empirical ansatz is found that
fits well to the Lempel-Ziv complexity of the MLZs for all lengths up to $10^6$
symbols.
