This paper proposes a low-complexity algorithm to calculate log likelihood ratios (LLRs) of coded bits, which is necessary for channel decoding in coded MIMO mobile communications. An approximate LLR needs to find a pair of transmitted signal candidates that can maximize the log likelihood function under a constraint that a coded bit is equal to either one or zero. The proposed algorithm can find such a pair simultaneously, whereas conventional ones find them individually. Specifically, the proposed method searches for such candidates in directions of the noise enhancement using the MMSE detection as a starting point. First, an inverse matrix which the MMSE weight matrix includes is obtained and then the power method derives eigenvectors of the inverse matrix as the directions of the noise enhancement. With some eigenvectors, one-dimensional search and hard decision are performed. From the resultant signals, the transmitted signal candidates to be required are selected on the basis of the log likelihood function. Computer simulations with 44 MIMO-OFDM, 16QAM, and convolutional codes (rate =1/2, 2/3) demonstrate that the proposed algorithm requires only 1.0 dB more Eb/N0 than that of the maximum likelihood detection (MLD) in order to achieve packet error rate of 10-3, while reducing the complexity to about 0.2% of that of MLD.