  A coded distributed computing (CDC) system aims to reduce the communication
load in the MapReduce framework. Such a system has $K$ nodes, $N$ input files,
and $Q$ Reduce functions. Each input file is mapped by $r$ nodes and each
Reduce function is computed by $s$ nodes. The objective is to achieve the
maximum multicast gain. There are known CDC schemes that achieve optimal
communication load. In some prominent known schemes, however, $N$ and $Q$ grow
too fast in terms of $K$, greatly reducing their gains in practical scenarios.
To mitigate the situation, some asymptotically optimal cascaded CDC schemes
with $r=s$ have been proposed by using symmetric designs. In this paper, we put
forward new asymptotically optimal cascaded CDC schemes with $r=s$ by using
$1$-designs. Compared with earlier schemes from symmetric designs, ours have
much smaller computation loads while keeping the other relevant parameters the
same. We also obtain new asymptotically optimal cascaded CDC schemes with more
flexible parameters compared with previously best-performing schemes.
