Many channels, such as wireless, magnetic recording, and jammed channels, tend to suffer from time intervals during which their reliability deteriorates significantly, to a degree that compromises data integrity. In some scenarios, receivers are able to detect the presence of these time intervals and may choose, accordingly, to “erase” some (or all of the) symbols received during such intervals. This technique causes symbol losses at known locations. This chapter is devoted to LDPC codes for correcting (or recovering) transmitted symbols that have been erased, called erasures. The simplest channel model with erasures is the binary erasure channel over which a transmitted bit is either correctly received or erased. There are two basic types of binary erasure channel, random and burst. Over a random binary erasure channel (BEC), erasures occur at random locations, each with the same probability of occurrence, whereas over a binary burst erasure channel (BBEC), erasures cluster into bursts. In this chapter, we first show that the LDPC codes constructed in Chapters 10–12, besides performing well over the AWGN channel, also perform well over the BEC. Then, we construct LDPC codes for correcting bursts of erasures. A list of references on LDPC codes for the binary erasure channels is given at the end of this chapter.

Iterative Decoding of LDPC Codes for the BEC

For transmission over the BEC, a transmitted symbol, 0 or 1, is either correctly received with probability 1 – p or erased with probability p, called the erasure probability, as shown in Figure 13.1.