Quasi-Cross Lattice Tilings with Applications to Flash Memory

We consider lattice tilings of ${\BBR}^{n}$ 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 of positive error and negative error (the ratio of which 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. We construct infinite families of perfect codes for any rational balance ratio, and provide a specific construction for $(2, 1, n)$-lattice tiling. The constructions are related to group splitting and modular $B_{1}$ sequences. We also study bounds on the parameters of lattice-tilings by quasi-crosses, connecting 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 is the only ones possible.