  We consider the problem of coding for the substring channel, in which
information strings are observed only through their (multisets of) substrings.
Because of applications to DNA-based data storage, due to DNA sequencing
techniques, 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 \emph{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. 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.
