NAND Flash memories are widely used as data storages today. The memories are not intrinsically error free because they are affected by several physical disturbances. Technology scaling and introduction of multi-level cell (MLC) has improved data density, but it has made error effect more significant. Error control codes (ECC) are essential to improve reliability of NAND Flash memories. Efficiency of codes depends on error characteristic of systems, and codes are required to be designed to reflect this characteristic. In MLC Flash memories, errors tend to direct values to neighborhood. These errors are a class of M -ary asymmetric symbol error. Some codes which reflect the asymmetric property were proposed. They are designed to correct only 1 level shift errors because almost all of the errors in the memories are in such errors. But technology scaling, increase of program/erase (P/E) cycles, and MLC storing the large number of bits can cause multiple-level shift. This paper proposes single error control codes which can correct an error of more than 1 levels shift. Because the number of levels to be corrected is selectable, we can fit it into noise magnitude. Furthermore, it is possible to add error detecting function for error of the larger shift. Proposed codes are equivalent to a conventional integer codes, which can correct 1 level shift, on a certain parameter. Therefore, the codes are said to be generalization of conventional integer codes. Evaluation results show information lengths to respective check symbol lengths are larger than nonbinary Hamming codes and other M -ary asymmetric symbol error correcting codes.
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