Efficient, nearly optimum addressing schemes based on partitioning the constellation into the union of blocks

Two efficient addressing schemes for the nearly optimum shaping of multi-dimensional signal spaces are introduced. Using K concentric circles, the 2-D subspaces are partitioned into shells of equal volume. The 2-D shells are indexed in the radial direction from zero to K-1. The average energy of a 2-D shell is proportional to its index plus a fixed offset. In an N = 2n-D space, Kⁿ shaping clusters of equal volume are obtained. Shaping is achieved by selecting T ⩽ Kⁿ of the N-D clusters with the least average energy (least sum of the 2-D indices). This results in a set of T integer n-tuples with components in the range [0, K-1] and the sum of the components being at most a given number L_max. The problem of addressing is to find a one-to-one mapping between the set of these n-tuples and the set of the integers [0, T - 1] such that the mapping and its inverse can be easily implemented. In the proposed schemes, the N-D clusters are grouped into blocks such that the addressing within the blocks, which is achieved using a common algorithm for all the blocks, has a low complexity