  This paper introduces a new family of reconstruction codes which is motivated
by applications in DNA data storage and sequencing. In such applications, DNA
strands are sequenced by reading some subset of their substrings. While
previous works considered two extreme cases in which all substrings of
pre-defined lengths are read or substrings are read with no overlap for the
single string case, this work studies two extensions of this paradigm. The
first extension considers the setup in which consecutive substrings are read
with some given minimum overlap. First, an upper bound is provided on the
attainable rates of codes that guarantee unique reconstruction. Then, efficient
constructions of codes that asymptotically meet that upper bound are presented.
In the second extension, we study the setup where multiple strings are
reconstructed together. Given the number of strings and their length, we first
derive a lower bound on the read substrings' length $\ell$ that is necessary
for the existence of multi-strand reconstruction codes with non-vanishing
rates. 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.
