Abstract
A unified approach to construct and optimize codes determined by their sparse parity-check matrices is presented. Parity-check matrices of quasi-cyclic (QC) LDPC block codes are obtained by replacing the nonzero elements of a base (seed) matrix by circulants. Replacing the nonzero elements either by companion matrices of elements from a finite field GF(2m) or by a formal variable D gives parity-check matrices of binary images of nonbinary LDPC block codes and LDPC convolutional codes, respectively. A set of performance measures applicable to different classes of LDPC codes are considered and a greedy algorithm for code performance optimization is presented. For a few classes of LDPC codes examples of codes combining good error-correcting performance with compact representation are obtained. Moreover, a specific channel model can easily be embedded into the optimization loop. Thereby, the code can be optimized for a specific channel. The efficiency of such an optimization is demonstrated via an example of Faster Than Nyquist (FTN) signaling using LDPC codes.
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