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A modified version of the sphere detection algorithm has been recently proposed to compute the bit likelihood values in MIMO systems. The proposed approach has an affordable computational cost, even for a large number of antennas, and is able to produce values very close to the actual likelihoods. This complexity reduction makes it possible to successfully utilize bit-interleaved coded modulation schemes based on iterative decoding (BICM-ID). In this paper, we propose to use a concatenation of two systematic low-density generator matrix (LDGM) codes as the channel code of the BICM-ID system. We show the ability of this affordable-complexity scheme to operate at SNRs near the Shannon capacity limit of the MIMO channel.