On Lattice Tilings by Symmetric Limited Magnitude Error Balls: Q-polynomials and Ideal Characterization

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,,Yp-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≡49 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≡89 and n≡2,46. Their sizes are also estimated.