General classification of lattice tilings for limited-magnitude two-coordinate error balls

Determine the existence of lattice tilings of \(\mathbb{Z}^n\) by the limited-magnitude error balls \(\mathcal{B}(n,2,k_1,k_2)\) for the general parameter ranges not resolved by the paper’s complete classifications and sufficiently-large-dimension nonexistence theorem.

Background

The paper completely classifies lattice tilings by $\mathcal{B}(n,2,3,0)$, proves nonexistence for the family $\mathcal{B}(n,2,k,k-1)$ with $k\geq2$, and establishes nonexistence in sufficiently large dimensions when $k_1+k_2+1$ is composite. It explicitly acknowledges that these results do not settle all remaining parameter choices. In particular, the unresolved general cases include parameter regimes outside the stated theorems, with the prime case $k_1+k_2+1=p$ identified later as especially difficult.