A technique for designing low-density parity-check (LDPC) error correcting codes for use with the partial-response channels commonly used in magnetic recording is presented. This technique combines the well-known density evolution method of Richardson and Urbanke for analyzing the performance of the LDPC decoder with a newly developed method for doing density evolution analysis of the Bahl-Cocke-Jelinek-Raviv (BCJR) channel decoder to predict the performance of LDPC codes in systems that employ both LDPC and BCJR decoders, and to search for good codes. We present examples of codes that perform 0.3dB to 0.5dB better than the regular column weight three codes employed in previous work.

A new algorithm is also presented, which we call "MTR enforcement". Typical magnetic recording systems employ not just an error correcting code, but also some form of run-length-limited code or maximum-transition-run (MTR) code. The MTR enforcement algorithm allows us to exploit the added redundancy imposed by the MTR code to increase performance over that of a magnetic recording system which does not employ the MTR enforcer. We show a gain of approximately 0.5dB from the MTR enforcer in a typical magnetic recording system. We also discuss methods of doing so-called "soft-error estimates", which attempt to extrapolate the bit-error-rate (BER) curve from Monte Carlo simulations down below the limits for which the traditional BER results are valid. The recent work by Yedidia on generalizations of the belief propagation algorithm is discussed, and we consider problems that arise in using this generalized belief propagation method for decoding LDPC codes.