  The study of constant-weight codes in $\ell_1$-metric was motivated by the
duplication-correcting problem for data storage in live DNA. It is interesting
to determine the maximum size of a code given the length $n$, weight $w$,
minimum distance $d$ and the alphabet size $q$. In this paper, based on graph
decompositions, we determine the maximum size of ternary codes with constant
weight $w$ and distance $2w-2$ for all sufficiently large length $n$.
Previously, this was known only for a very sparse family $n$ of density
$4/w(w-1)$.
