Constant L₁-Weight Codes Under L∞-Metric

• Published in: IEEE transactions on information theory Vol. 70; no. 7; pp. 4928 - 4945
• Main Authors: Chen, Tingting, He, Zhiwen, Ge, Gennian
• Format: Journal Article
• Language: English
• Published: IEEE 01.07.2024
• Subjects: Additives; Chebyshev approximation; Codes; constant L₁-weight codes; error control; Germanium; graphs; Insertion and deletion codes; L∞-metric; Measurement; Symbols; Vectors
• ISSN: 0018-9448
• DOI: 10.1109/TIT.2024.3385419

This paper studies the construction of constant <inline-formula> <tex-math notation="LaTeX">L_{1} </tex-math></inline-formula>-weight codes under <inline-formula> <tex-math notation="LaTeX">L_{\infty } </tex-math></inline-formula>-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 <inline-formula> <tex-math notation="LaTeX">w+1 </tex-math></inline-formula> are also provided. In general, for codes with distance D, length <inline-formula> <tex-math notation="LaTeX">w+1 </tex-math></inline-formula>, and weight <inline-formula> <tex-math notation="LaTeX">tD+r </tex-math></inline-formula>, we derive the size of the largest code by using Hall's theorem for integers <inline-formula> <tex-math notation="LaTeX">r=0 </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">t\in [{0,3}] </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">w\geq t-1 </tex-math></inline-formula>. When <inline-formula> <tex-math notation="LaTeX">r=\frac {D}{2} </tex-math></inline-formula> with D being even, codes with a cardinality that surpasses the known existing sizes by the additive term <inline-formula> <tex-math notation="LaTeX">\left \lceil{ \frac {1}{3}\binom {t+1}{2}}\right \rceil \left \lfloor{ \frac {w+1}{3}}\right \rfloor </tex-math></inline-formula> are constructed based on the structure of group divisible designs in combinatorics.