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.

Integer codes correct errors of a given type, which means that for a given communication channel and modulator we can choose the type of the errors (which are the most common) then construct integer code capable of correcting those errors. A new general construction of single (1) error correctable integer codes will be presented. Comparison between single and multiple (1) error correctable integer codes over AWGN channel using QAM scheme will be presented.

Integer codes are very flexible and can be applied in different modulation schemes. A soft decoding algorithm for integer codes will be introduced. Comparison of symbol error probability (SEP) versus signal-to-noise ratio (SNR) between soft and hard decoding using integer coded modulation shows us that we can obtain at least 2 dB coding gain. Also, we shall compare our results with trellis coded modulation (TCM) because of their similar decoding schemes and complexity.

In this paper we present the exact expressions for the bit error probability over a Gaussian noise channel of coded QAM using single error correcting integer codes. It is shown that the proposed integer codes have a better performance with respect to the lower on the bit error probability for trellis coded modulation.

In this paper, we investigate the problem how to construct integer codes capable of correcting any single error in the set {1,t,...,tk-1} and generalize our results to obtain (e1,e2,...,es) single error correctable codes where ei's are different elements in A. Moreover, we shall give the exact form of the check matrix in most of the classes considered in this paper.