I. Introduction
Multilevel coding (MLC), proposed by Imai and Hirakawa in 1977 [1], is a powerful coded modulation scheme capable of achieving both bandwidth- and power-efficient communication [2], [3]. The key idea behind the MLC is to protect the individual bits using different binary codes and use -ary signal constellation (for more details, the reader is referred to [1]–[9]). The decoding is based on so-called multistage decoding (MSD) algorithm, in which decisions from prior (lower) decoding stages are passed to the next (higher) stages (see [4]). Despite its attractiveness with respect to large coding gain, the MLC with the MSD algorithm has serious limitation for use in high-speed applications, such as optical communications, which is due to the inherently large delay of the MSD algorithm. One possible solution is to use the parallel independent decoding (PID) [3]. In the MLC/PID scheme, the information bit stream is split into different levels, and the corresponding bits at different levels are encoded using different encoders, and then combined into a signal point using appropriate mapping rule. At the receiver side, decoders at different levels operate independently and in parallel. For more details on MLC in general, an interesting reader is referred to [4], and for more details on PID to [3]. It has been widely recognized that the use of Gray mapping and PID at each level separately, with optimally chosen component codes [3], [5], [6], may lead to channel capacity approaching performance. Our idea is to use low-density parity-check (LDPC) codes as component codes. LDPC codes have been shown to be able to achieve impressive coding gains on a variety of channels, including the fiber-optic communication channel [10], [11]. Furthermore, their low-complexity decoding algorithm makes them a good choice for combining with the low-delay PID scheme.