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.

A novel multilevel construction for permutation codes is presented. A permutation code of length n is a subset of all the vectors obtained from coordinate permutations on the vector (0,1,. . . ,n-1). We would like to construct a permutation code with cardinality as large as possible for a given code length n and a minimum distance. The proposed construction is available when n = 2m (m is a positive integer). We exploit m-constant weight binary codes as component codes and combine them in a multilevel way. Permutation codes with various parameters can be constructed by selecting appropriate combination of component codes. Furthermore, multi-stage decoding is available for decoding the permutation codes constructed by the proposed construction.