Constant L₁-Weight Codes Under L∞-Metric

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.