2025/01/15 by Zhihao Guan, Guan, Zhihao, Hengjia Wei +2
Computer Science · Materials Science · Mathematics · #Cellular Automata and Applications #Combinatorics (math.CO) #FOS: Mathematics #Quasicrystal Structures and Properties #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2501.08636
openalex publication_date 2025/01/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Limited-magnitude errors modify a transmitted integer vector in at most t entries, where each entry can increase by at most \kp or decrease by at most \km. This channel model is particularly relevant to applications such as flash memories and DNA storage. A perfect code for this channel is equivalent to a tiling of \Zn by asymmetric limited-magnitude balls \cB(n,t,\kp,\km). In this paper, we focus on the case where t=2 and \km=\kp-1, and we derive necessary conditions on m and n for the existence of a lattice tiling of \cB(n,2,m,m-1). Specifically, we prove that if such a tiling exists, then either 4≤ m ≤ 512 and n<7.23m+4, or m>512 and n<4m. In particular, for m=2 and m=3, we show that no lattice tiling of \cB(n,2,2,1) or \cB(n,2,3,2) exists for any n≥ 3.