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

Abstract

In this paper, we study the lattice tilings of $\mathbb{Z}^n$ by the symmetric limited magnitude error balls $\mathcal{B}(n,t,k,k)$, 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 $\mathcal{B}(n,t,k,k)$. Then, an ideal-theoretic characterization of the lattice tiling condition is developed: for an odd prime $p$ dividing $|\mathcal{B}(n,t,k,k)|$ exactly once, a lattice tiling would force a certain polynomial to lie outside a certain ideal of $\mathbb{F}_p[Y_1,\dots,Y_{\frac{p-1}{2}}]$. Applying this criterion yields congruence conditions on the dimension $n$ that rule out lattice tilings; for $2\le t\le10$ and $1\le k\le10$, a systematic Maple computation over all primes $3\le p\le29$ 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 $k\equiv4\pmod9$ and the base-$3$ addition of $t$ and $n-1-t$ produces exactly one carry; and no lattice tiling of $\mathbb{Z}^n$ by $\mathcal{B}(n,n-1,k,k)$ exists if $k\equiv8\pmod9$ and $n\equiv2,4\pmod6$. Their sizes are also estimated.