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We explicitly construct space-time codes for a multiple access system composed of K users transmitting simultaneously to the base station in block-fading channels. Each user encodes its own data independently without cooperation between users. Our codes design uses the idea of linear dispersion encoding together with circulant matrices assigned to each user that allow low-complexity linear minimum mean square error (MMSE) decoding when the channel realization is known to the base station. Moreover, the ergodic sum rate of the proposed design is studied and compared with the sum capacity.