On Lattice Tilings by Symmetric Limited Magnitude Error Balls: Q-polynomials and Ideal Characterization
Abstract
In this paper, we study the lattice tilings of ℤn by the symmetric limited magnitude error balls ℬ(n,t,k,k), which play a crucial role in flash memory applications. Using generating functions and complete Bell polynomials, we derive an explicit formula for the r-th Q-polynomial associated with ℬ(n,t,k,k). Then, an ideal-theoretic characterization of the lattice tiling condition is developed: for an odd prime p dividing |ℬ(n,t,k,k)| exactly once, a lattice tiling would force a certain polynomial to lie outside a certain ideal of 𝔽p[Y1,…,Y(p−1)/2]. Applying this criterion yields congruence conditions on the dimension n that rule out lattice tilings; for 2≤ t≤10 and 1≤ k≤10, a systematic Maple computation over all primes 3≤ p≤29 excludes, on average, 35.65% of all dimensions, with a maximum of 76.69%. Finally, we obtain two infinite families of parameters (n,t,k) for which no lattice tiling exists: no lattice tiling exists if k≡4pmod9 and the base-$3$ addition of t and n−1−t produces exactly one carry; and no lattice tiling of ℤn by ℬ(n,n−1,k,k) exists if k≡8pmod9 and n≡2,4pmod6. Their sizes are also estimated.
Related
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BibTeX
@misc{xiao2026lattice,
title = {{On Lattice Tilings by Symmetric Limited Magnitude Error Balls: Q-polynomials and Ideal Characterization}},
author = {Ang Xiao},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02394},
eprint = {2611.02394},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02394},
}
RIS
TY - UNPB
TI - On Lattice Tilings by Symmetric Limited Magnitude Error Balls: Q-polynomials and Ideal Characterization
AU - Ang Xiao
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.02394
UR - https://arxiv.org/abs/2611.02394
AB - In this paper, we study the lattice tilings of ℤ^n by the symmetric limited magnitude error balls ℬ(n,t,k,k), which play a crucial role in flash memory applications. Using generating functions and complete Bell polynomials, we derive an explicit formula for the r-th Q-polynomial associated with ℬ(n,t,k,k). Then, an ideal-theoretic characterization of the lattice tiling condition is developed: for an odd prime p dividing |ℬ(n,t,k,k)| exactly once, a lattice tiling would force a certain polynomial to lie outside a certain ideal of 𝔽_p[Y_1,…,Y_(p−1)/2]. Applying this criterion yields congruence conditions on the dimension n that rule out lattice tilings; for 2≤ t≤10 and 1≤ k≤10, a systematic Maple computation over all primes 3≤ p≤29 excludes, on average, 35.65% of all dimensions, with a maximum of 76.69%. Finally, we obtain two infinite families of parameters (n,t,k) for which no lattice tiling exists: no lattice tiling exists if k≡4pmod9 and the base-$3$ addition of t and n−1−t produces exactly one carry; and no lattice tiling of ℤ^n by ℬ(n,n−1,k,k) exists if k≡8pmod9 and n≡2,4pmod6. Their sizes are also estimated.
ER -