Rotationally invariant nonlinear trellis codes for two-dimensionalmodulation

A general parity-check equation is presented that defines rotationally invariant trellis codes of rate k/(k+1) for two-dimensional signal sets. This parity-check equation is used to find rate k/(k+1) codes for 4PSK, 8PSK, 16PSK, and QAM signal sets by systematic code searches. The MPSK codes exhibit smaller free Euclidean distances than nonrotationally invariant linear codes with the same number of states. However, since the nonlinear codes have a smaller number of nearest neighbors, their performance at moderate signal to noise ratios is close to that of the best linear codes. The rotationally invariant QAM codes with 8, 32, 64, and 256 states achieve the same free Euclidean distance as the best linear codes. Transparency of user information under phase rotations is accomplished either by conventional differential encoding and decoding, or by integrating this function directly into the code trellis