New rotationally invariant four-dimensional trellis codes

Two new classes of rotationally invariant trellis codes are constructed. A simple method is used to check the rotational invariance of a given code in the process of searching for optimum trellis codes. A class of linear trellis codes with constraint lengths 2-9 using four-dimensional constellations is presented. Simulation results show that a 180/spl deg/ rotationally invariant, constraint length 8, linear trellis code achieves about 0.4-dB real coding gain compared to the best constraint length 6 code, while the trellis complexity is only four times that of the constraint length 6 code. On the other hand, the constraint length 6 code that has been adopted for use in the V.34 28.8-kbit/s modem standard has the same 0.4-dB real coding gain compared to a constraint length 4 code which has also been adopted for the standard, but it requires sixteen times the trellis complexity. A class of fully rotationally invariant nonlinear trellis codes with constraint lengths 6-11 is also presented. Simulation results show that a 90/spl deg/ rotationally invariant, constraint length 8, nonlinear trellis code performs almost as well as the best linear code.