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Low density generator matrix (LDGM) codes are a particular class of low density parity check (LDPC) codes with very low encoding complexity. Single LDGM codes present high error-floors, which can be substantially reduced with the serial concatenation of two LDGM (SCLDGM) codes. We propose a technique to obtain good SCLDGM codes using extrinsic information transfer (EXIT) functions in a novel way. Although the optimization is performed for AWGN channels with binary signaling, the resulting codes are also optimal for AWGN and perfectly-interleaved Rayleigh fading channels with non-binary signaling and perfect CSI at reception, provided that Gray mapping is utilized. Optimized regular and irregular SCLDGM codes outperform heuristically-designed LDGM codes existing in the literature, and have a performance similar to or better than that of irregular repeat accumulate (IRA) codes.