  In this paper, we consider a multiple-input-multiple-output optical wireless
communication (MIMO-OWC) system suffering from log-normal fading. In this
scenario, a general criterion for the design of full large-scale diversity
space code (FLDSC) with the maximum likelihood (ML) detector is developed.
Based on our criterion, FLDSC is attained if and only if all the entries of the
space coding matrix are positive. Particularly for $2\times 2$ MIMO-OWC with
unipolar pulse amplitude modulation (PAM), a closed-form linear FLDSC
satisfying this criterion is attained by smartly taking advantage of some
available properties as well as by developing some new interesting properties
on Farey sequences in number theory to rigorously attack the continuous and
discrete variables mixed max-min problem. In fact, this specific design not
only proves that a repetition code (RC) is the best linear FLDSC, but also
uncovers a significant difference between MIMO radio frequency (RF)
communications and MIMO-OWC that space-only transmission is sufficient for a
full diversity achievement. Computer simulations demonstrate that FLDSC
substantially outperforms spatial multiplexing with the same total optical
power and spectral efficiency and the latter obtains only the small-scale
diversity gain.
