  We consider the problem of coding for the substring channel, in which
information strings are observed only through their (multisets of) substrings.
Due to existing DNA sequencing techniques and applications in DNA-based storage
systems, interest in this channel has renewed in recent years. In contrast to
existing literature, we consider a noisy channel model where information is
subject to noise before its substrings are sampled, motivated by in-vivo
storage. We study two separate noise models, substitutions or deletions. In
both cases, we examine families of codes which may be utilized for
error-correction and present combinatorial bounds on their sizes. Through a
generalization of the concept of repeat-free strings, we show that the added
required redundancy due to this imperfect observation assumption is sublinear,
either when the fraction of errors in the observed substring length is
sufficiently small, or when that length is sufficiently long. This suggests
that no asymptotic cost in rate is incurred by this channel model in these
cases. Moreover, we develop an efficient encoder for such constrained strings
in some cases. Finally, we show how a similar encoder can be used to avoid
formation of secondary-structures in coded DNA strands, even when accounting
for imperfect structures.
