Methods and apparatus for use in communication systems using recursive modulation schemes with a Low Density Generator Matrix code (including an irregular repeat accumulate (IRA) code) are described that have reduced complexity and thus reduced cost compared to prior art systems. A communication system is described in which the transmitter concatenates a low density generator matrix code with an accumulator followed by a recursive modulator in order to eliminate the use of an interleaver, and in which the receiver combines the decoder for the accumulator and the soft demodulator into a single joint decoder in order reduce the number of components and complexity. Another variation is also described in which the transmitter is further simplified by eliminated the accumulator altogether, and in which the receiver is further simplified by replacing the joint decoder with a soft demodulator prior to the LDGM soft decoder.