In this paper, we investigate Tanner's lower bound for the minimum distance of regular LDPC codes based on combinatorial designs. We first determine Tanner's lower bound for LDPC codes which are defined by modifying bipartite graphs obtained from combinatorial designs known as Steiner systems. Then we show that Tanner's lower bound agrees with or exceeds conventional lower bounds including the BCH bound, and gives the true minimum distance for some EG-LDPC codes.

In this letter, we show the effectiveness of a double-loop algorithm based on the concave-convex procedure (CCCP) in decoding linear codes. For this purpose, we numerically compare the error performance of CCCP-based decoding algorithm with that of a conventional iterative decoding algorithm based on belief propagation (BP). We also investigate computational complexity and its relation to the error performance.

In this paper, an iterative decoding algorithm for channels with additive linear dynamical noise is presented. The proposed algorithm is based on the tightly coupled two inference algorithms: the sum-product algorithm which infers the information symbols of an low density parity check (LDPC) code and the Kalman smoothing algorithm which infers the channel states. The linear dynamical noise are the noise generated from a linear dynamical system. We often encounter such noise (i.e., additive colored noise) in practical communication and storage systems. The conventional iterative decoding algorithms such as the sum-product algorithm cannot derive full potential of turbo codes nor LDPC codes over such a channel because the conventional algorithms are designed under the independence assumption on the noise. Several simulations have been performed to assess the performance of the proposed algorithm. From the simulation results, it can be concluded that the Kalman smoothing algorithm deserves to be implemented in a decoder when the linear dynamical part of the linear dynamical noise is dominant rather than the white Gaussian noise part. In such a case, the performance of the proposed algorithm is far superior to that of the conventional algorithm.

We propose a scheme for the design of irregular low-density parity-check (LDPC) codes based on Euclidian Geometry using Latin square matrices of random sequence. Our scheme is a deterministic method that allows the easy design of good irregular LDPC codes for any code rate and degree distribution. We optimize the LDPC codes using the Gaussian approximation method. A Euclidean Geometry LDPC code (EG-LDPC) is used as the basis for the construction of an irregular LDPC code. The base EG-LDPC code is extended by splitting rows and columns using a table of Latin square matrices of random sequence to determine the edges along which to split. We provide simulation results for codes constructed in this manner evaluated in terms of bit error rate (BER) performance in AWGN channels. We believe that our scheme is superior in terms of computational requirements and resulting BER performance in comparison to creation of irregular LDPC codes by means of random construction using a search algorithm to exclude cycles of length four.

A parity check matrix for a binary linear code defines a bipartite graph (Tanner graph) which is isomorphic to a subgraph of a factor graph which explains a mechanism of the iterative decoding based on the sum-product algorithm. It is known that this decoding algorithm well approximates MAP decoding, but degradation of the approximation becomes serious when there exist cycles of short length, especially length 4, in Tanner graph. In this paper, based on the generating idempotents, we propose some methods to design parity check matrices for cyclic codes which define Tanner graphs with no cycles of length 4. We also show numerically error performance of cyclic codes by the iterative decoding implemented on factor graphs derived from the proposed parity check matrices.

Gallager has defined an ensemble of regular low density parity check (LDPC) codes for deriving the ensemble performance of regular LDPC codes. The ensemble is called the Gallager ensemble. In this paper, we define a new ensemble of LDPC codes, called extended Gallager ensemble, which is a natural extension of the Gallager ensemble. It is shown that an extended Gallager ensemble has potential to achieve larger typical minimum distance ratio than that of the original Gallager ensemble. In particular, the extended Gallager ensembles based on the Hamming and extended Hamming codes have typical minimum distance ratio which is very close to the asymptotic Gilbert-Varshamov bound. Furthermore, decoding performance of an instance of an extended Gallager ensemble, called an extended LDPC code, has been examined by simulation. The results show good block error performance of extended LDPC codes.

A coded modulation scheme based on a low density parity check (LDPC) code is presented. A modified sum-product algorithm suitable for the LDPC-coded modulation scheme is also devised. Several simulation results show the excellent decoding performance of the proposed coding scheme. For example, an LDPC-coded 8PSK scheme of block length 3976 symbols achieves the symbol error probability 10-5 at only 1.2 dB away from the Shannon limit of the channel.