Designing optimum 2D-TCM schemes using new systematic convolutional encoders over GF (2N)

A new family of recursive systematic convolutional (RSC) encoders operating over Galois field GF (2^N) using the left-circulate (LCIRC) function is introduced for trellis-coded modulation (TCM) transmissions. As compared with the previously proposed encoders using the LCIRC function, the novelty of the present encoder is given by the systematic output, for any encoding rate. Therefore, we found a simple encoder output expression, using GF (2^N) operators, such that the binary representation of the encoder output symbol includes explicitly the input symbol´s binary representation. Simulations were run to test the symbol error rate (SER) performances of the TCM transmission, using the RSC-LCIRC encoders, over additive white Gausssian noise (AWGN) channel for 8-phase shift keying (PSK) and 16-quadrature amplitude modulation (QAM). Also, we simulated the corresponding optimum binary encoders known in the literature. The simulation SER results confirm the optimality of the new RSC-LCIRC encoders.