  Coded distributed computing (CDC) introduced by Li \emph{et al.} can greatly
reduce the communication load for MapReduce computing systems. In the general
cascaded CDC with $K$ workers, $N$ input files and $Q$ Reduce functions, each
input file will be mapped by $r$ workers and each Reduce function will be
computed by $s$ workers such that coding techniques can be applied to achieve
the maximum multicast gain. The main drawback of most existing CDC schemes is
that they require the original data to be split into a large number of input
files that grows exponentially with $K$, which can significantly increase the
coding complexity and degrade system performance. In this paper, we first use a
classic combinatorial structure $t$-design, for any integer $t\geq 2$, to
develop a low-complexity and asymptotically optimal CDC with $r=s$. The main
advantages of our scheme via $t$-design are two-fold: 1) having much smaller
$N$ and $Q$ than the existing schemes under the same parameters $K$, $r$ and
$s$; and 2) achieving smaller communication loads compared with the
state-of-the-art schemes. Remarkably, unlike the previous schemes that realize
on large operation fields, our scheme operates on the minimum binary field
$\mathbb{F}_2$. Furthermore, we show that our construction method can
incorporate the other combinatorial structures that have a similar property to
$t$-design. For instance, we use $t$-GDD to obtain another asymptotically
optimal CDC scheme over $\mathbb{F}_2$ that has different parameters from
$t$-design. Finally, we show that our construction method can also be used to
construct CDC schemes with $r\neq s$ that have small file number and Reduce
function number.
