This paper proposes a fountain coding system which has lower decoding erasure rate and lower space complexity of the decoding algorithm than the Raptor coding systems. A main idea of the proposed fountain code is employing shift and exclusive OR to generate the output packets. This technique is known as the zigzag decodable code, which is efficiently decoded by the zigzag decoder. In other words, we propose a fountain code based on the zigzag decodable code in this paper. Moreover, we analyze the overhead, decoding erasure rate, decoding complexity, and asymptotic overhead of the proposed fountain code. As a result, we show that the proposed fountain code outperforms the Raptor codes in terms of the overhead and decoding erasure rate. Simulation results show that the proposed fountain coding system outperforms Raptor coding system in terms of the overhead and the space complexity of decoding.

This paper derives the average symbol and bit weight distributions for the irregular non-binary cluster low-density parity-check (LDPC) code ensembles. Moreover, we give the exponential growth rates of the average weight distributions in the limit of large code length. We show the condition that the typical minimum distances linearly grow with the code length.

This paper constructs non-binary codes correcting a single b-burst of insertions or deletions with large cardinalities. This paper also provides insertion and deletion correcting algorithms of the constructed codes and evaluates a lower bound of the cardinalities of the constructed codes. Moreover, we evaluate a non-asymptotic upper bound on the cardinalities of arbitrary codes which correct a single b-burst of insertions or deletions.

This paper constructs packet-oriented erasure correcting codes and their systematic forms for the distributed storage systems. The proposed codes are encoded by exclusive OR and bit-level shift operation. By the shift operation, the encoded packets are slightly longer than the source packets. This paper evaluates the extra length of the encoded packets, called overhead, and shows that the proposed codes have smaller overheads than the zigzag decodable codes, which are existing codes using bit-level shift operation and exclusive OR.

In this paper, we investigate the error floors of non-binary low-density parity-check (LDPC) codes transmitted over the memoryless binary-input output-symmetric (MBIOS) channels. We provide a necessary and sufficient condition for successful decoding of zigzag cycle codes over the MBIOS channel by the belief propagation decoder. We consider an expurgated ensemble of non-binary LDPC codes by using the above necessary and sufficient condition, and hence exhibit lower error floors. Finally, we show lower bounds of the error floors for the expurgated LDPC code ensembles over the MBIOS channels.

The fixed points of the belief propagation decoder for non-binary low-density parity-check (LDPC) codes are referred to as stopping constellations. In this paper, we give the stopping constellation distributions for the irregular non-binary LDPC code ensembles defined over the general linear group. Moreover, we derive the exponential growth rate of the average stopping constellation distributions in the limit of large codelength.

In this paper, we investigate the error floors of the non-binary low-density parity-check codes transmitted over the binary erasure channels under belief propagation decoding. We propose a method to improve the decoding erasure rates in the error floors by optimizing labels in zigzag cycles in the Tanner graphs of codes. Furthermore, we give lower bounds on the bit and the symbol erasure rates in the error floors. The simulation results show that the presented lower bounds are tight for the codes designed by the proposed method.

Luby et al. derived evolution of degree distributions in residual graphs for irregular LDPC code ensembles. Evolution of degree distributions in residual graphs is important characteristic which is used for finite-length analysis of the expected block and bit error probability over the binary erasure channel. In this paper, we derive detailed evolution of degree distributions in residual graphs for irregular LDPC code ensembles with joint degree distributions.

We propose a digital image watermarking method satisfying information hiding criteria (IHC) for robustness against JPEG compression, cropping, scaling, and rotation. When a stego-image is cropped, the marking positions of watermarks are unclear. To detect the position in a cropped stego-image, a marker or synchronization code is embedded with the watermarks in a lattice pattern. Attacks by JPEG compression, scaling, and rotation cause errors in extracted watermarks. Against such errors, the same watermarks are repeatedly embedded in several areas. The number of errors in the extracted watermarks can be reduced by using a weighted majority voting (WMV) algorithm. To correct residual errors in output of the WMV algorithm, we use a high-performance error-correcting code: a low-density parity-check (LDPC) code constructed by progressive edge-growth (PEG). In computer simulations using the IHC ver. 4 the proposed method could a bit error rate of 0, the average PSNR was 41.136 dB, and the computational time for synchronization recovery was less than 10 seconds. The proposed method can thus provide high image quality and fast synchronization recovery.

Fountain codes are erasure correcting codes realizing reliable communication systems for the multicast on the Internet. The zigzag decodable fountain (ZDF) codes are one of generalization of the Raptor codes, i.e., applying shift operation to generate the output packets. The ZDF codes are decoded by a two-stage iterative decoding algorithm, which combines the packet-wise peeling algorithm and the bit-wise peeling algorithm. By the bit-wise peeling algorithm and shift operation, ZDF codes outperform Raptor codes under iterative decoding in terms of decoding erasure rates and overheads. However, the bit-wise peeling algorithm spends long decoding time. This paper proposes fast bit-wise decoding algorithms for the ZDF codes. Simulation results show that the proposed algorithm drastically reduces the decoding time compared with the previous algorithm.

In this paper, we compare the decoding error rates in the error floors for non-binary low-density parity-check (LDPC) codes over general linear groups with those for non-binary LDPC codes over finite fields transmitted through the q-ary memoryless symmetric channels under belief propagation decoding. To analyze non-binary LDPC codes defined over both the general linear group GL(m, F2) and the finite field F2m, we investigate non-binary LDPC codes defined over GL(m3, F2m4). We propose a method to lower the error floors for non-binary LDPC codes. In this analysis, we see that the non-binary LDPC codes constructed by our proposed method defined over general linear group have the same decoding performance in the error floors as those defined over finite field. The non-binary LDPC codes defined over general linear group have more choices of the labels on the edges which satisfy the condition for the optimization.

Low-density parity-check (LDPC) codes are widely used in communication systems for their high error-correcting performance. This survey introduces the elements of LDPC codes: decoding algorithms, code construction, encoding algorithms, and several classes of LDPC codes.

A random access scheme is a fundamental scenario in which the users transmit through a shared channel and cannot coordinate with each other. Recently, successive interference cancellation (SIC) is introduced into the random access scheme. The SIC decodes the transmitted packets using collided packets. The coded slotted ALOHA (CSA) is a random access scheme using the SIC. The CSA encodes each packet by a local code prior to transmission. It is known that the CSA achieves excellent throughput. On the other hand, it is reported that shift operation improves the decoding performance for packet-oriented erasure correcting coding systems. In this paper, we propose a protocol which applies the shift operation to the CSA. Numerical simulations show that the proposed protocol achieves better throughput and packet loss rate than the CSA. Moreover, we analyze the asymptotic behavior of the throughput and the decoding erasure rate for the proposed protocol by the density evolution.

Maximum run-length limited codes are constraint codes used in communication and data storage systems. Insertion/deletion correcting codes correct insertion or deletion errors caused in transmitted sequences and are used for combating synchronization errors. This paper investigates the maximum run-length limited single insertion/deletion correcting (RLL-SIDC) codes. More precisely, we construct efficiently encodable and decodable RLL-SIDC codes. Moreover, we present its encoding and decoding algorithms and show the redundancy of the code.

In this paper, we propose a message passing decoding algorithm which lowers decoding error rates in the error floor regions for non-binary low-density parity-check (LDPC) codes transmitted over the binary erasure channel (BEC) and the memoryless binary-input output-symmetric (MBIOS) channels. In the case for the BEC, this decoding algorithm is a combination with belief propagation (BP) decoding and maximum a posteriori (MAP) decoding on zigzag cycles, which cause decoding errors in the error floor region. We show that MAP decoding on the zigzag cycles is realized by means of a message passing algorithm. Moreover, we extend this decoding algorithm to the MBIOS channels. Simulation results demonstrate that the decoding error rates in the error floor regions by the proposed decoding algorithm are lower than those by the BP decoder.