Abstract-This paper investigates decoding of low-density parity-check (LDPC) codes over the binary erasure channel (BEC). We study the iterative and maximum-likelihood (ML) decoding of LDPC codes on this channel. We derive bounds on the ML decoding of LDPC codes on the BEC. We then present an improved decoding algorithm. The proposed algorithm has almost the same complexity as the standard iterative decoding. However, it has better performance. Simulations show that we can decrease the error rate by several orders of magnitude using the proposed algorithm. We also provide some graph-theoretic properties of different decoding algorithms of LDPC codes over the BEC which we think are useful to better understand the LDPC decoding methods, in particular, for finite-length codes.

This paper investigates properties of polar codes that can be potentially useful in real-world applications. We start with analyzing the performance of finite-length polar codes over the binary erasure channel (BEC), while assuming belief propagation as the decoding method. We provide a stopping set analysis for the factor graph of polar codes, where we find the size of the minimum stopping set. We also find the girth of the graph for polar codes. Our analysis along with bit error rate (BER) simulations demonstrate that finite-length polar codes show superior error floor performance compared to the conventional capacity-approaching coding techniques. In order to take advantage from this property while avoiding the shortcomings of polar codes, we consider the idea of combining polar codes with other coding schemes. We propose a polar code-based concatenated scheme to be used in Optical Transport Networks (OTNs) as a potential real-world application. Comparing against conventional concatenation techniques for OTNs, we show that the proposed scheme outperforms the existing methods by closing the gap to the capacity while avoiding error floor, and maintaining a low complexity at the same time.

In this paper, we study polar codes from a practical point of view. In particular, we study concatenated polar codes and rate-compatible polar codes. First, we propose a concatenation scheme including polar codes and Low-Density Parity-Check (LDPC) codes. We will show that our proposed scheme outperforms conventional concatenation schemes formed by LDPC and Reed-Solomon (RS) codes. We then study two rate-compatible coding schemes using polar codes. We will see that polar codes can be designed as universally capacity achieving rate-compatible codes over a set of physically degraded channels. We also study the effect of puncturing on polar codes to design rate-compatible codes.

Abstract-This paper first introduces an improved decoding algorithm for low-density parity-check (LDPC) codes over binaryinput-output-symmetric memoryless channels. Then some fundamental properties of punctured LDPC codes are presented. It is proved that for any ensemble of LDPC codes, there exists a puncturing threshold. It is then proved that for any rates R1 and R2 satisfying 0 < R1 < R2 < 1, there exists an ensemble of LDPC codes with the following property. The ensemble can be punctured from rate R1 to R2 resulting in asymptotically good codes for all rates R1 R R2. Specifically, this implies that rates arbitrarily close to one are achievable via puncturing. Bounds on the performance of punctured LDPC codes are also presented. It is also shown that punctured LDPC codes are as good as ordinary LDPC codes. For BEC and arbitrary positive numbers R 1 < R 2 < 1, the existence of the sequences of punctured LDPC codes that are capacity-achieving for all rates R 1 R R 2 is shown. Based on the above observations, a method is proposed to design good punctured LDPC codes over a broad range of rates. Finally, it is shown that the results of this paper may be used for the proof of the existence of the capacity-achieving LDPC codes over binary-input-outputsymmetric memoryless channels.

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