In this letter, we propose a new modification to the belief propagation (BP) decoding algorithm for Finite-Geometry low-density parity-check (LDPC) codes. The modification is based on introducing feedback into the iterative process, which can break the oscillations of bit log-likelihood ratio (LLR) values. Simulations show that, with a given maximum iteration, the "feedback BP" (FBP) algorithm can achieve better performance than the conventional belief propagation algorithm.

The most powerful channel coding schemes, namely those based on turbo codes and low-density parity-check (LDPC) Gallager codes, have in common the principle of iterative decoding. However, the relative coding structures and decoding algorithms are substantially different. This paper presents a 2048-bit, rate-1/2 soft decision decoder for a new class of codes known as Turbo Gallager Codes. These codes are turbo codes with properly chosen component convolutional codes such that they can be successfully decoded by means of the decoding algorithm used for LDPC codes, i.e., the belief propagation algorithm working on the code Tanner graph. These coding schemes are important in practical terms for two reasons: (i) they can be encoded as classical turbo codes, giving a solution to the encoding problem of LDPC codes; (ii) they can also be decoded in a fully parallel manner, partially overcoming the routing congestion bottleneck of parallel decoder VLSI implementations thanks to the locality of the interconnections. The implemented decoder can support up to 1 Gbit/s data rate and performs up to 48 decoding iterations ensuring both high throughput and good coding gain. In order to evaluate the performance and the gate complexity of the decoder VLSI architecture, it has been synthesized in a 0.18 µm standard-cell CMOS technology.

The Reliability-Based Hybrid ARQ (RB-HARQ) scheme, which can be used with error correcting codes using soft-input soft-output (SISO) decoders such as convolutional codes and turbo codes has been proposed. In the RB-HARQ scheme, the error rate performance is improved by selecting the retransmission bits based on Log Likelihood Ratio (LLR) of each bit in the receiver. However, the receiver has to send the bit positions of retransmission bits to the transmitter. Therefore, the RB-HARQ scheme requires a great number of feedback bits. On the other hand, Low Density Parity Check (LDPC) codes are attracting a lot of interest, recently. Because LDPC codes can achieve near Shannon limit performance and be decoded easily compared to turbo code. In this paper, we evaluate the RB-HARQ scheme using LDPC code. Moreover, we propose a RB-HARQ scheme that requires a fewer feedback bits by utilizing a code structure of LDPC code. We refer to the scheme as the RB-HARQ (row base) scheme. We show that the RB-HARQ and RB-HARQ (row base) schemes using LDPC code have better error rate performance than the scheme without ARQ. We also show that the RB-HARQ (row base) scheme has a good trade-off between error rate performance and the number of feedback bits compared to the RB-HARQ scheme.

Density evolution has recently been used to analyze the iterative decoding of Low Density Parity Check (LDPC) codes, Turbo codes, and Serially Concatenated Convolutional Codes (SCCC). The density evolution technique makes it possible to explain many characteristics of iterative decoding including convergence of performance and preferred structures for the constituent codes. While the analytic density evolution methods were applied to LDPC codes, the simulation based density evolution methods were used for Turbo codes and SCCC due to analytic difficulties. In this paper, several density evolution ideas in the literature are used to analyze common code structures and it is shown that those ideas yield consistent results. In order to do that, we derive expressions for density evolution of SCCC with a simple 2-state constituent code. The analytic expressions are based on the sum-product and min-sum algorithms, and the thresholds are evaluated for both message passing algorithms. Particularly, for the min-sum algorithm, the density evolution with Gaussian approximation is derived and used to analyze the effect of scaling soft information. The scaling of extrinsic information slows down the convergence of soft information or avoids an overestimation effect of it and results in better performance, and its gain is maximized in particular constituent codes. Similar approaches are made for LDPC code. We show that the scaling gain is noticeable in the LDPC code as well. This scaling gain is analyzed with both density evolution and simulation performance. The expected scaling gain by density evolution matches well with the achievable scaling gain from simulation results. These results can be extended to the irregular LDPC codes based on the degree distribution for the min-sum algorithm. All density evolution algorithms used in this paper are based on the Gaussian approximation for the exchanged messages.