  Motivated by applications to DNA storage, we study reconstruction and
list-reconstruction schemes for integer vectors that suffer from
limited-magnitude errors. We characterize the asymptotic size of the
intersection of error balls in relation to the code's minimum distance. We also
devise efficient reconstruction algorithms for various limited-magnitude error
parameter ranges. We then extend these algorithms to the list-reconstruction
scheme, and show the trade-off between the asymptotic list size and the number
of required channel outputs. These results apply to all codes, without any
assumptions on the code structure. Finally, we also study linear reconstruction
codes with small intersection, as well as show a connection to
list-reconstruction codes for the tandem-duplication channel.
