  In this paper, we consider the outer channel for DNA-based data storage,
where each DNA string is either correctly transmitted, or being erased, or
being corrupted by uniformly distributed random substitution errors, and all
strings are randomly shuffled with each other. We first derive the capacity of
the outer channel, which surprisingly implies that the uniformly distributed
random substitution errors are only as harmful as the erasure errors. Next, we
propose efficient coding schemes which encode the bits at the same position of
different strings into a codeword. We compute the soft/hard information of each
bit, which allows us to independently decode the bits within a codeword,
leading to an independent decoding scheme. To improve the decoding performance,
we measure the reliability of each string based on the independent decoding
result, and perform a further step of decoding over the most reliable strings,
leading to a joint decoding scheme. Simulations with low-density parity-check
codes confirm that the joint decoding scheme can reduce the frame error rate by
more than 3 orders of magnitude compared to the independent decoding scheme,
and it can outperform the state-of-the-art decoding scheme in the literature in
a wide parameter regions.
