  Motivated by the duplication-correcting problem for data storage in live DNA,
we study the construction of constant-weight codes in $\ell_1$-metric. By using
packings and group divisible designs in combinatorial design theory, we give
constructions of optimal codes over non-negative integers and optimal ternary
codes with $\ell_1$-weight $w\leq 4$ for all possible distances. In general, we
derive the size of the largest ternary code with constant weight $w$ and
distance $2w-2$ for sufficiently large length $n$ satisfying $n\equiv
1,w,-w+2,-2w+3\pmod{w(w-1)}$.
