Constant L₁-Weight Codes Under L∞-Metric - Chen, Tingting; He, Zhiwen; Ge, Gennian (IEEE transactions on information theory, 2024)
This paper studies the construction of constant L_{1} -weight codes under L_{∞ } -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≥ 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.