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.