  This paper studies reconstruction of strings based upon their substrings
spectrum. Under this paradigm, it is assumed that all substrings of some fixed
length are received and the goal is to reconstruct the string. While many
existing works assumed that substrings are received error free, we follow in
this paper the noisy setup of this problem that was first studied by Gabrys and
Milenkovic. The goal of this study is twofold. First we study the setup in
which not all substrings in the multispectrum are received, and then we focus
on the case where the read substrings are not error free. In each case we
provide specific code constructions of strings that their reconstruction is
guaranteed even in the presence of failure in either model. We present
efficient encoding and decoding maps and analyze the cardinality of the code
constructions, while studying the cases where the rates of our codes approach
1.
