Spatially Coupled LDPC Codes Constructed from Protographs

In this paper, we construct protograph-based spatially coupled low-density parity-check (SC-LDPC) codes by coupling together a series of L disjoint, or uncoupled, LDPC code Tanner graphs into a single coupled chain. By varying L, we obtain a flexible family of code ensembles with varying rates and frame lengths that can share the same encoding and decoding architecture for arbitrary L. We demonstrate that the resulting codes combine the best features of optimized irregular and regular codes in one design: capacity approaching iterative belief propagation (BP) decoding thresholds and linear growth of minimum distance with block length. In particular, we show that, for sufficiently large L, the BP thresholds on both the binary erasure channel (BEC) and the binary-input additive white Gaussian noise channel (AWGNC) saturate to a particular value significantly better than the BP decoding threshold and numerically indistinguishable from the optimal maximum a-posteriori (MAP) decoding threshold of the uncoupled LDPC code. When all variable nodes in the coupled chain have degree greater than two, asymptotically the error probability converges at least doubly exponentially with decoding iterations and we obtain sequences of asymptotically good LDPC codes with fast convergence rates and BP thresholds close to the Shannon limit. Further, the gap to capacity decreases as the density of the graph increases, opening up a new way to construct capacity achieving codes on memoryless binary-input symmetric-output (MBS) channels with low-complexity BP decoding.Comment: Submitted to the IEEE Transactions on Information Theor

Two generalizations of ideal matrices and their applications

In this paper, two kinds of generalizations of ideal matrices, generalized ideal matrices and double ideal matrices. are obtained and studied, The concepts of generalized ideal matrices and double ideal matrices are proposed, and their ranks and maxima.linearly independent groups are verified.The initial motivation to study double cyclic matrices is to study the quasi cyclic codes of the fractional index. In this paper, the generalized form of the quasi cyclic codes, i.e. the {\phi}-quasi cyclic codes. and the construction of the generated matrix are given by the double ideal matrix

Artin's primitive root conjecture -a survey -

This is an expanded version of a write-up of a talk given in the fall of 2000 in Oberwolfach. A large part of it is intended to be understandable by non-number theorists with a mathematical background. The talk covered some of the history, results and ideas connected with Artin's celebrated primitive root conjecture dating from 1927. In the update several new results established after 2000 are also discussed.Comment: 87 pages, 512 references, to appear in Integer

Mixed Precision Multi-frame Parallel Low-Density Parity-Check Code Decoder

As the demand for high speed and high quality connectivity is increasing exponentially, channels are getting more and more crowded. The need for a high performance and low error floor channel decoder is apparent. Low-density parity-check code (LDPC) is a linear error correction code that can reach near Shannon limit. In this work, LDPC code construction and decoding algorithms are discussed, the LDPC decoder, in fully parallel and partial parallel, was implemented, and the features and issues related to corresponding architecture are analyzed. Furthermore, a multi-frame processing approach, based on pipelining and out-of-order processing, is proposed. The implemented decoder achieves 12.6 Gbps at 3.0 dB SNR. The mixed precision scheme is explored by adding precision control and alignment units before and after check node units (CNU) to improve performance, as well as error floor. By mixing the 6-bit and 5-bit precision CNUs at 1:1 ratio, the decoder reaches ~0.5 dB lower FER and BER while retaining a low error floor

Mixed Precision Multi-frame Parallel Low-Density Parity-Check Code Decoder

As the demand for high speed and high quality connectivity is increasing exponentially, channels are getting more and more crowded. The need for a high performance and low error floor channel decoder is apparent. Low-density parity-check code (LDPC) is a linear error correction code that can reach near Shannon limit. In this work, LDPC code construction and decoding algorithms are discussed, the LDPC decoder, in fully parallel and partial parallel, was implemented, and the features and issues related to corresponding architecture are analyzed. Furthermore, a multi-frame processing approach, based on pipelining and out-of-order processing, is proposed. The implemented decoder achieves 12.6 Gbps at 3.0 dB SNR. The mixed precision scheme is explored by adding precision control and alignment units before and after check node units (CNU) to improve performance, as well as error floor. By mixing the 6-bit and 5-bit precision CNUs at 1:1 ratio, the decoder reaches ~0.5 dB lower FER and BER while retaining a low error floor