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, in Motahari et al., (2013), 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.
