Conditions on the optimality of multilevel codes

The key point in designing a multilevel coding (MLC) scheme is the proper assignment of code rates to the individual coding levels. Assuming low complex multistage decoding (MSD), at each level i of a MLC scheme an equivalent channel can be defined for transmission of binary symbol xⁱ. If the rates Rⁱ at the individual coding levels are chosen to be equal to the capacities Cⁱ of the equivalent channels, MLC together with MSD is optimum in the sense of capacity. In this paper, we present more general conditions on the individual rates Rⁱ for MLC to be optimal, when overall maximum likelihood decoding (MLD) instead of MSD is used. To maximize the minimum distance of the Euclidean space code, balancing the products d_i²·δ_i for all levels i has often been proposed as a design rule. Here, d_i denotes the minimum intra subset Euclidean distance at the ith partitioning level and δ_i the minimum Hamming distance of component code Cⁱ. We show that MLC schemes which are designed by this balanced distances rule (BDR) can achieve capacity only if an extremely complex overall MLD is employed