Conditions on the optimality of multilevel codes

Abstract

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 x/sup i/. If the rates R/sup i/ at the individual coding levels are chosen to be equal to the capacities C/sup i/ 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 R/sup i/ 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/sub i//sup 2//spl middot//spl delta//sub i/ for all levels i has often been proposed as a design rule. Here, d/sub i/ denotes the minimum intra subset Euclidean distance at the ith partitioning level and /spl delta//sub i/ the minimum Hamming distance of component code C/sup i/. 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.