High code rate, low-density parity-check codes with guaranteed minimum distance and stopping weight

Recent schemes have been proposed for magnetic recording that combine a maximum a posteriori probability channel detector for a partial response channel and a low-density parity-check (LDPC) code. The signal processing for these schemes uses the message passing algorithm for iteratively sharing information between the channel detector and the LDPC decoder. At high signal to noise ratios, the sector-error probability of a scheme using an LDPC code is related to the minimum distance of the code. Because these schemes use an LDPC code whose minimum distance has not been characterized, the sector-error probability of these schemes also has not been characterized for high signal to noise ratios.; We show that an analysis of isomorphic classes of binary matrices results in new design criteria for an LDPC code with a guaranteed minimum distance. We present a new scheme that uses an LDPC code synthesized using these design criteria. We also present synthesis algorithms that select columns of a parity-check matrix in a random manner resulting in an LDPC code with a minimum distance of at least eight. We show that search algorithms found short block length or high code rate, LDPC codes using either a synthesis algorithm that uses random selection with replacement or a synthesis algorithm that uses random selection without replacement. By using a truncation of the union bound, we show that the new scheme meets the standard sector-error probability requirement for disk drives. We also show via computer simulation that, for low signal to noise ratios, the new scheme has a low sector-error probability.; Finally, we present new design criteria for a class of LDPC codes that apply to the binary erasure channel and partial response systems. We define an LDPC code with a stopping weight W as an LDPC code described by a parity-check matrix that contains a set of W columns (with special properties). We show that search algorithms using the new design criteria found short block length or high code rate LDPC codes with a guaranteed minimum stopping weight.