Quasi-cross lattice tilings with applications to flash memory

We consider lattice tilings of ℝ ⁿ by a shape we call a (k ₊, k _-,n)-quasi-cross. Such lattices form perfect error-correcting codes which correct a single limited-magnitude error with prescribed maximal-magnitudes k ₊ and k _- of positive error and negative error respectively (the ratio of which, β = k _-/k ₊, is called the balance ratio). These codes can be used to correct both disturb and retention errors in flash memories, which are characterized by having limited magnitudes and different signs. For any rational 0 < β < 1 we construct an infinite family of (k ₊, k _-, n)-quasi-cross lattice tilings with balance ratio k _-/k ₊= β. We also provide a specific construction for an infinite family of (2, 1, n)-quasi-cross lattice tilings. The constructions are related to group splitting and modular B ₁ sequences. In addition, we study bounds on the parameters of lattice-tilings by quasi-crosses, and express them in terms of the arm lengths of the quasi-crosses and the dimension. We also prove constraints on group splitting, a specific case of which shows that the parameters of the lattice tiling by (2,1,n)-quasi-crosses are the only ones possible for these quasi-crosses.