  This paper tackles two problems that are relevant to coding for insertions
and deletions. These problems are motivated by several applications, among them
is reconstructing strands in DNA-based storage systems. Under this paradigm, a
word is transmitted over some fixed number of identical independent channels
and the goal of the decoder is to output the transmitted word or some close
approximation of it. The first part of this paper studies the deletion channel
that deletes a symbol with some fixed probability $p$, while focusing on two
instances of this channel. Since operating the maximum likelihood (ML) decoder
in this case is computationally unfeasible, we study a slightly degraded
version of this decoder for two channels and its expected normalized distance.
We identify the dominant error patterns and based on these observations, it is
derived that the expected normalized distance of the degraded ML decoder is
roughly $\frac{3q-1}{q-1}p^2$, when the transmitted word is any $q$-ary
sequence and $p$ is the channel's deletion probability. We also study the cases
when the transmitted word belongs to the Varshamov Tenengolts (VT) code or the
shifted VT code. Additionally, the insertion channel is studied as well as the
case of two insertion channels. These theoretical results are verified by
corresponding simulations. The second part of the paper studies optimal
decoding for a special case of the deletion channel, the $k$-deletion channel,
which deletes exactly $k$ symbols of the transmitted word uniformly at random.
In this part, the goal is to understand how an optimal decoder operates in
order to minimize the expected normalized distance. A full characterization of
an efficient optimal decoder for this setup, referred to as the maximum
likelihood* (ML*) decoder, is given for a channel that deletes one or two
symbols.
