  Decoding sequences that stem from multiple transmissions of a codeword over
an insertion, deletion, and substitution channel is a critical component of
efficient deoxyribonucleic acid (DNA) data storage systems. In this paper, we
consider a concatenated coding scheme with an outer nonbinary low-density
parity-check code or a polar code and either an inner convolutional code or a
time-varying block code. We propose two novel decoding algorithms for inference
from multiple received sequences, both combining the inner code and channel to
a joint hidden Markov model to infer symbolwise a posteriori probabilities
(APPs). The first decoder computes the exact APPs by jointly decoding the
received sequences, whereas the second decoder approximates the APPs by
combining the results of separately decoded received sequences and has a
complexity that is linear with the number of sequences. Using the proposed
algorithms, we evaluate the performance of decoding multiple received sequences
by means of achievable information rates and Monte-Carlo simulations. We show
significant performance gains compared to a single received sequence. In
addition, we succeed in improving the performance of the aforementioned coding
scheme by optimizing both the inner and outer codes.
