  Multiply constant-weight codes (MCWCs) were introduced recently to improve
the reliability of certain physically unclonable function response. In this
paper, the bounds of MCWCs and the constructions of optimal MCWCs are studied.
Firstly, we derive three different types of upper bounds which improve the
Johnson-type bounds given by Chee {\sl et al.} in some parameters. The
asymptotic lower bound of MCWCs is also examined. Then we obtain the asymptotic
existence of two classes of optimal MCWCs, which shows that the Johnson-type
bounds for MCWCs with distances $2\sum_{i=1}^mw_i-2$ or $2mw-w$ are
asymptotically exact. Finally, we construct a class of optimal MCWCs with total
weight four and distance six by establishing the connection between such MCWCs
and a new kind of combinatorial structures. As a consequence, the maximum sizes
of MCWCs with total weight less than or equal to four are determined almost
completely.
