In this paper, we propose demodulators for the Golden and Alamouti codes in amplify-and-forward (AF) cooperative communication with one relay. The proposed demodulators output exact log likelihood ratios (LLRs) with recursion based on the Jacobian logarithm. The cooperative system with the proposed demodulator for the Golden code has the benefit of efficient data transmission, while the system for the Alamouti code has low demodulation complexity. Quantitative analyses of computational complexity of the proposed demodulators are conducted. The transmission performance for various relay location and power settings is evaluated on cooperative orthogonal frequency division multiplexing (OFDM)-based wireless local area network (LAN) systems. In evaluations, the optimal relay location and power settings are found. The cooperative system with the proposed demodulators for the Golden and Alamouti codes offers 1.5 and 1.9 times larger areas where 10.8 and 5.4Mbit/s can be obtained than a non-cooperative (direct) system in a typical office environment, respectively.