Iterative decoding of linear block codes: A parity-check orthogonalization approach

It is widely accepted that short cycles in Tanner graphs deteriorate the performance of message-passing algorithms. This discourages the use of these algorithms on Tanner graphs (TGs) of well-known algebraic codes such as Hamming codes, Bose-Chaudhuri-Hocquenghem codes, and Reeddolomon codes. Yedidia et al. presented a method to generate code representations suitable for message-passing algorithms. This method does not guarantee a representation free of four-cycles. In this correspondence, we present an algorithm to convert an arbitrary linear block into a code with orthogonal parity-check equations. A combinatorial argument is used to prove that the algorithm guarantees a four-cycle free representation for any linear code. The effects of removing four-cycles on the performance of a belief propagation decoder for the binary erasure channel are studied in detail by analyzing the structures in different representations. Finally, we present bit-error rate (BER) and block-error rate (BLER) performance curves of linear block codes under belief propagation algorithms for the binary erasure channel and the additive white Gaussian noise (AWGN) channel in order to demonstrate the improvement in performance achieved with the help of the proposed algorithm.