  The problem of string reconstruction based on its substrings spectrum has
received significant attention recently due to its applicability to DNA data
storage and sequencing. In contrast to previous works, we consider in this
paper a setup of this problem where multiple strings are reconstructed
together. Given a multiset $S$ of strings, all their substrings of some fixed
length $\ell$, defined as the $\ell$-profile of $S$, are received and the goal
is to reconstruct all strings in $S$. A multi-strand $\ell$-reconstruction code
is a set of multisets such that every element $S$ can be reconstructed from its
$\ell$-profile. Given the number of strings~$k$ and their length~$n$, we first
find a lower bound on the value of $\ell$ necessary for existence of
multi-strand $\ell$-reconstruction codes with non-vanishing asymptotic rate. We
then present two constructions of such codes and show that their rates
approach~$1$ for values of $\ell$ that asymptotically behave like the lower
bound.
