  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 \emph{all} substrings of
some fixed length are read or substrings are read with no overlap, this work
considers the setup in which consecutive substrings are read with some given
minimum overlap. First, upper bounds are provided on the attainable rates of
codes that guarantee unique reconstruction. Then, we present efficient
constructions of asymptotically optimal codes that meet the upper bound.
