  The errors occurring in DNA-based storage are correlated in nature, which is
a direct consequence of the synthesis and sequencing processes. In this paper,
we consider the memory-$k$ nanopore channel model recently introduced by Hamoum
et al., which models the inherent memory of the channel. We derive the maximum
a posteriori (MAP) decoder for this channel model. The derived MAP decoder
allows us to compute achievable information rates for the true DNA storage
channel assuming a mismatched decoder matched to the memory-$k$ nanopore
channel model, and quantify the loss in performance assuming a small memory
length--and hence limited decoding complexity. Furthermore, the derived MAP
decoder can be used to design error-correcting codes tailored to the DNA
storage channel. We show that a concatenated coding scheme with an outer
low-density parity-check code and an inner convolutional code yields excellent
performance.
