This paper studies the construction of constant$L_{1}$-weight codes under$L_{\infty }$-metric, which could be used to design codes addressing 0-indels (i.e., the insertion/deletion of 0’s only). Based on the patterns of small codes, we determine the maximum size for codes with distance 2 and length 3 for any constant weight, and several constructions and algorithms for suboptimal codes with general length$w+1$are also provided. In general, for codes with distance D, length$w+1$, and weight$tD+r$, we derive the size of the largest code by using Hall’s theorem for integers$r=0$,$t\in [{0,3}]$and$w\geq t-1$. When$r=\frac {D}{2}$with D being even, codes with a cardinality that surpasses the known existing sizes by the additive term$\left \lceil{ \frac {1}{3}\binom {t+1}{2}}\right \rceil \left \lfloor{ \frac {w+1}{3}}\right \rfloor $are constructed based on the structure of group divisible designs in combinatorics.