  A connected graph $G$ is called strongly Menger edge connected if $G$ has
min\{deg$_G(x)$, deg$_G(y)$\} edge-disjoint paths between any two distinct
vertices $x$ and $y$ in $G$. In this paper, we consider two types of strongly
Menger edge connectivity of the line graphs of $n$-dimensional hypercube-like
networks with faulty edges, namely the $m$-edge-fault-tolerant and
$m$-conditional edge-fault-tolerant strongly Menger edge connectivity. We show
that the line graph of any $n$-dimensional hypercube-like network is
$(2n-4)$-edge-fault-tolerant strongly Menger edge connected for $n\geq 3$ and
$(4n-10)$-conditional edge-fault-tolerant strongly Menger edge connected for
$n\geq 4$. The two bounds for the maximum number of faulty edges are best
possible.
