  Most DNA sequencing technologies are based on the shotgun paradigm: many
short reads are obtained from random unknown locations in the DNA sequence. A
fundamental question, studied in arXiv:1203.6233, is what read length and
coverage depth (i.e., the total number of reads) are needed to guarantee
reliable sequence reconstruction. Motivated by DNA-based storage, we study the
coded version of this problem;i.e., the scenario where the DNA molecule being
sequenced is a codeword from a predefined codebook. Our main result is an exact
characterization of the capacity of the resulting shotgun sequencing channel as
a function of the read length and coverage depth. In particular, our results
imply that, while in the uncoded case, $O(n)$ reads of length greater than
$2\log{n}$ are needed for reliable reconstruction of a length-$n$ binary
sequence, in the coded case, only $O(n/\log{n})$ reads of length greater than
$\log{n}$ are needed for the capacity to be arbitrarily close to $1$.
