Signal space diversity (SSD) introduced by Boutros and Viterbo tremendously improves the error performance over fading channels without any power or bandwidth sacrifice. Maximum benefit is obtained with full diversity (FD) multidimensional codebooks and the rather complex maximum likelihood (ML) detection. In the present work, we propose a generalized combinatorial representation and an associated low-complexity list-based detection algorithm that works for any non-full or full diversity multidimensional codebook. The algorithm includes three options of lists offering different complexity-performance trade-offs. One of the lists has random size and is guaranteed to contain the ML point, while the other two lists have fixed or limited size. Some analytical results are presented for the list sizes. The algorithm also includes a smart search to find the most probable codebook point in the list. Numerical results show that the proposed algorithm yields optimal or close-to-optimum performance with a noteworthy detection complexity reduction. Therefore, its adoption can be of practical interest, especially in applications where powerful error-correcting codes are not supported.