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 $k_{+}$ and $k_{-}$ of positive error and negative error respectively (the ratio of which, $\beta=k_{-}/k_{+}$, 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