Iterative coding methods for the binary symmetric channel and magnetic recording channel

The primary concern of this thesis is to develop better coding methods for two systems of practical importance: the magnetic recording system, and the fiber-optic communication channel. For the magnetic recording system, we consider low-density parity-check (LDPC) codes. LDPC codes, introduced by Gallager in the 60s, are randomly constructed linear block codes whose parity-check matrices have a very low density of 1s. LDPC codes can be decoded using an iterative message-passing decoder, which is an approximate version of the maximum-likelihood decoder. The following capacity result can be proven for LDPC codes: For noise levels below a threshold level, arbitrarily low bit-error rates can be obtained using LDPC codes of large enough block-length and performing high enough iterations of message-passing decoding. In this thesis, we develop a low-complexity method to estimate the threshold of LDPC codes over channels that are used to model the magnetic recording system. We develop heuristics for constructing LDPC codes for magnetic recording read channels. A fiber-optic communication channel is modeled as a binary symmetric channel (BSC). The high rate of transmission (1–10 Gbps) in optical channels implies that the coding method must be implementable at very high speeds (at least 600–1000 Mbps with very few logic gates). At present, Reed-Solomon (RS) codes are the most popular codes for BSCs such as fiber-optic channels. In this thesis, we propose a new system based on majority-logic decoding of finite geometry codes. We show that the new method works better than competing RS codes, and involves lesser number of bit operations to implement than competing RS codes.