In this paper, trellis coded modulation (TCM) schemes are designed using recursive encoders over Galois field GF(2^N). These encoders are designed using the nonlinear left-circulate (LCIRC) function. The LCIRC function performs a bit left circulation over the representation word. Different encoding rates are obtained for these encoders when using different representation word lengths at the input and the output, denoted as Nin and N, respectively. A generalized 1-delay GF(2^N) recursive encoder scheme using LCIRC is proposed for performance analysis and optimization, for any possible encoding rate, Nin/N. Several TCM schemes using pulse amplitude modulation (PAM), and phase shift keying (PSK) are considered. The minimum Euclidian distance is estimated for all schemes and a general expression is found as a function of the word lengths Nin and N. The symbol error rate (SER) is estimated by simulation for PSKTCM transmissions over an additive white Gaussian noise (AWGN) channel.

Keywords: GF(2^N) encoders, Left-circulate function, Euclidian distance, Trelliscoded modulation.