Finite Geometry LDPC Research Articles

Globally Coupled Finite Geometry and Finite Field LDPC Coding Schemes

This paper presents two types of concatenated LDPC coding schemes which are viewed as <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">generalized globally coupled</i> ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">GC</i> ) <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">LDPC</i> coding schemes in which outer codes serve as the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">local codes</i> for correcting local errors and inner codes serve as <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">global coupling codes</i> to correct global errors. The first type of concatenated LDPC coding scheme globally couples a finite geometry (FG) LDPC code as the local code and a finite field (FF) LDPC code as the global coupling code. This type of global coupling, called <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">GC-FG/FF-LDPC coupling</i> , combines the distinct features of both FG- and FF-LDPC codes to achieve low error rates at a rapid decoding convergence and an error performance close to the Shannon limit. Decoding of a GC-FG/FF-LDPC code is carried out in <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">two iterative phases</i> , global/local or local/global. In the second type of concatenated LDPC coding scheme, both local and global coupling codes are FF-LDPC codes. If both local and global coupling codes are constructed from the same finite field and have the same graphical structures, a GC-FF/FF-LDPC code can be decoded in one phase or two phases iteratively, otherwise, it can be decoded in two phases. Construction of GC-FF/FF-LDPC codes is very flexible in lengths and rates. The proposed two-phase iterative decoding is practically implementable.

High‐throughput Bit Flipping decoder for structured LDPC codes

Low-density parity-check (LDPC) codes are predominantly used in many energy scavenging devices, data centres, and communication devices. In this work, the authors propose a high-throughput parallel bit-flipping (BF) decoder using multithreshold BF algorithm. The decoder is endowed with features including low interconnect complexity, simpler computations, and high throughput. The decoder has been synthesised in 65 nm technology for (273, 191) and (1023, 781) finite-geometry LDPC (FG-LDPC) codes. These decoders require an area of 0 . 1 and 0 . 45 mm 2 , and they achieve an average throughput of 147 . 57 and 268 . 5 Gbps and energy efficiency of 0 . 92 and 0 . 91 pJ/bit for (273, 191) and (1023, 781) FG-LDPC codes, respectively. Compared to the state-of-the-art design, the proposed designs offer 4 . 5 times higher normalised throughput.

A construction of the high-rate regular quasi-cyclic LDPC codes

In this paper, a scheme to construct the high-rate regular quasi-cyclic low-density parity-check (QC-LDPC) codes is developed based on the finite geometry low-density parity-check (LDPC) codes. To achieve this, we first decompose the EG-LDPC code into a set of the block submatrices with each of them being cyclic. Utilizing the decomposed structure, we then construct an auxiliary matrix to facilitate us for identifying and eliminating 6-cycles of the block submatrices. In the simulation, seven regular QC-LDPC codes with code rates being 4/5, 7/8, 10/11, 11/12, and 12/13 of moderate code lengths are constructed with this scheme. The performances of these codes demonstrate the effectiveness of the proposed scheme.

Deterministic Constructions of Binary Measurement Matrices From Finite Geometry

Deterministic constructions of measurement matrices in compressed sensing (CS) are considered in this paper. The constructions are inspired by the recent discovery of Dimakis, Smarandache and Vontobel which says that parity-check matrices of good low-density parity-check (LDPC) codes can be used as provably good measurement matrices for compressed sensing under l1-minimization. The performance of the proposed binary measurement matrices is mainly theoretically analyzed with the help of the analyzing methods and results from (finite geometry) LDPC codes. Particularly, several lower bounds of the spark (i.e., the smallest number of columns that are linearly dependent, which totally characterizes the recovery performance of l0-minimization) of general binary matrices and finite geometry matrices are obtained and they improve the previously known results in most cases. Simulation results show that the proposed matrices perform comparably to, sometimes even better than, the corresponding Gaussian random matrices. Moreover, the proposed matrices are sparse, binary, and most of them have cyclic or quasi-cyclic structure, which will make the hardware realization convenient and easy.

Family of finite geometry low-density parity-check codes for quantum key expansion

We consider a quantum key expansion (QKE) protocol based on entanglement-assisted quantum error-correcting codes (EAQECCs). In these protocols, a seed of a previously shared secret key is used in the postprocessing stage of a standard quantum key distribution protocol like the Bennett-Brassard 1984 protocol, in order to produce a larger secret key. This protocol was proposed by Luo and Devetak, but codes leading to good performance have not been investigated. We look into a family of EAQECCs generated by classical finite geometry (FG) low-density parity-check (LDPC) codes, for which very efficient iterative decoders exist. A critical observation is that almost all errors in the resulting secret key result from uncorrectable block errors that can be detected by an additional syndrome check and an additional sampling step. Bad blocks can then be discarded. We make some changes to the original protocol to avoid the consumption of the preshared key when the protocol fails. This allows us to greatly reduce the bit error rate of the key at the cost of a minor reduction in the key production rate, but without increasing the consumption rate of the preshared key. We present numerical simulations for the family of FG LDPC codes, and show that this improved QKE protocol has a good net key production rate even at relatively high error rates, for appropriate choices of these codes.

Iterative Algorithms for Decoding a Class of Two-Step Majority-Logic Decodable Cyclic Codes

Codes constructed based on finite geometries form a large class of cyclic codes with large minimum distances which can be decoded with simple majority-logic decoding in one or multiple steps. In 2001, Kou, Lin and Fossorier showed that the one-step majority-logic decodable finite geometry codes form a class of cyclic LDPC codes whose Tanner graphs are free of cycles of length 4. These cyclic finite geometry LDPC codes perform very well over the AWGN channel using iterative decoding based on belief propagation (IDBP) and have very low error-floors. However, the standard IDBP is not effective for decoding other cyclic finite geometry codes because their Tanner graphs contain too many short cycles of length 4 which severely degrade the decoding performance. This paper investigates iterative decoding of two-step majority-logic decodable finite geometry codes. Three effective algorithms for decoding these codes are proposed. These algorithms are devised based on the orthogonal structure of the parity-check matrices of the codes to avoid or reduce the degrading effect of the short cycles of length 4. These decoding algorithms provide significant coding gains over the standard IDBP using either the sum-product or the min-sum algorithms.

Quasi-cyclic LDPC codes: an algebraic construction

This paper presents two new large classes of QC-LDPC codes, one binary and one non-binary. Codes in these two classes are constructed by array dispersions of row-distance constrained matrices formed based on additive subgroups of finite fields. Experimental results show that codes constructed perform very well over the AWGN channel with iterative decoding based on belief propagation. Codes of a subclass of the class of binary codes have large minimum distances comparable to finite geometry LDPC codes and they offer effective tradeoff between error performance and decoding complexity when decoded with low-complexity reliability-based iterative decoding algorithms such as binary message passing decoding algorithms. Non-binary codes decoded with a Fast-Fourier Transform based sum-product algorithm achieve significantly large coding gains over Reed-Solomon codes of the same lengths and rates decoded with either the hard-decision Berlekamp-Massey algorithm or the algebraic soft-decision Kotter-Vardy algorithm. They have potential to replace Reed-Solomon codes in some communication or storage systems where combinations of random and bursts of errors (or erasures) occur.

Hybrid decoding of finite geometry low-density parity-check codes

For finite geometry low-density parity-check codes, heavy row and column weights in their parity check matrix make the decoding with even min–Sum (MS) variants computationally expensive. To alleviate it, the authors present a class of hybrid schemes by concatenating a parallel bit flipping (BF) variant with an MS variant. Meanwhile, the BF variant, with much less computational complexity, attempts decoding the receive sequence firstly, and only decoding failure of the BF variant triggers the MS variant. Hence the BF variant performance heavily impacts the overall hybrid scheme, which is illustrated in two case studies. Computational and hardware complexity is then elaborated to justify the feasibility of the hybrid schemes. In most SNR region of interest, without compromising performance or convergence rate, the proposed hybrid schemes can save substantial computational complexity when compared with the MS variant decoding alone.

Asymmetric quantum codes: constructions, bounds and performance

Recently, quantum error-correcting codes have been proposed that capitalize on the fact that many physical error models lead to a significant asymmetry between the probabilities for bit- and phase-flip errors. An example for a channel that exhibits such asymmetry is the combined amplitude damping and dephasing channel, where the probabilities of bit and phase flips can be related to relaxation and dephasing time, respectively. We study asymmetric quantum codes that are obtained from the Calderbank–Shor–Steane (CSS) construction. For such codes, we derive upper bounds on the code parameters using linear programming. A central result of this paper is the explicit construction of some new families of asymmetric quantum stabilizer codes from pairs of nested classical codes. For instance, we derive asymmetric codes using a combination of Bose–Chaudhuri–Hocquenghem (BCH) and finite geometry low-density parity-check (LDPC) codes. We show that the asymmetric quantum codes offer two advantages, namely to allow a higher rate without sacrificing performance when compared with symmetric codes and vice versa to allow a higher performance when compared with symmetric codes of comparable rates. Our approach is based on a CSS construction that combines BCH and finite geometry LDPC codes.

An improved decoding algorithm for finite-geometry LDPC codes

In this letter, an improved bit-flipping decoding algorithm for high-rate finite-geometry low-density parity-check (FG-LDPC) codes is proposed. Both improvement in performance and reduction in decoding delay are observed by flipping multiple bits in each iteration. Our studies show that the proposed algorithm achieves an appealing tradeoff between performance and complexity for FG-LDPC codes.

Computation of Groebner Basis for Systematic Encoding of Generalized Quasi-Cyclic Codes

Generalized quasi-cyclic (GQC) codes form a wide and useful class of linear codes that includes thoroughly quasi-cyclic codes, finite geometry (FG) low density parity check (LDPC) codes, and Hermitian codes. Although it is known that the systematic encoding of GQC codes is equivalent to the division algorithm in the theory of Grobner basis of modules, there has been no algorithm that computes Grobner basis for all types of GQC codes. In this paper, we propose two algorithms to compute Grobner basis for GQC codes from their parity check matrices: echelon canonical form algorithm and transpose algorithm. Both algorithms require sufficiently small number of finite-field operations with the order of the third power of code-length. Each algorithm has its own characteristic; the first algorithm is composed of elementary methods, and the second algorithm is based on a novel formula and is faster than the first one for high-rate codes. Moreover, we show that a serial-in serial-out encoder architecture for FG LDPC codes is composed of linear feedback shift registers with the size of the linear order of code-length; to encode a binary codeword of length n, it takes less than 2n adder and 2n memory elements. Keywords: automorphism group, Buchberger's algorithm, division algorithm, circulant matrix, finite geometry low density parity check (LDPC) codes.

Improved parallel weighted bit-flipping decoding algorithm for LDPC codes

Aiming at seeking a low-complexity decoder with fast decoding convergence speed for short or medium low-density parity-check (LDPC) codes, an improved parallel weighted bit-flipping (IPWBF) algorithm, which is applied flexibly for two classes of codes is presented here. For LDPC codes with low column weight in their parity check matrix, both bootstrapping and loop detection procedures, described in the existing literature, are included in IPWBF. Furthermore, a novel delay-handling procedure is introduced to prevent the codeword bits of high reliability from being flipped too hastily. For large column weight finite geometry LDPC codes, only the delay-handling procedure is included in IPWBF to show its effectiveness. Extensive simulation results demonstrate that the proposed algorithm achieves a good tradeoff between performance and complexity.

On the Stopping Distance and Stopping Redundancy of Finite Geometry LDPC Codes

The stopping distance and stopping redundancy of a linear code are important concepts in the analysis of the performance and complexity of the code under iterative decoding on a binary erasure channel. In this paper, we studied the stopping distance and stopping redundancy of Finite Geometry LDPC (FG-LDPC) codes, and derived an upper bound of the stopping redundancy of FG-LDPC codes. It is shown from the bound that the stopping redundancy of the codes is less than the code length. Therefore, FG-LDPC codes give a good trade-off between the performance and complexity and hence are a very good choice for practical applications.

Hybrid Iterative Decoding for Low-Density Parity-Check Codes Based on Finite Geometries

In this letter, a two-stage hybrid iterative decoding algorithm which combines two iterative decoding algorithms is proposed to reduce the computational complexity of finite geometry low-density parity-check (FG-LDPC) codes. We introduce a fast weighted bit-flipping (WBF) decoding algorithm for the first stage decoding. If the first stage decoding fails, the decoding is continued by the powerful belief propagation (BP) algorithm. The proposed hybrid decoding algorithm greatly reduces the computational complexity while maintains the same performance compared to that of using the BP algorithm only.

On the stopping distance of finite geometry ldpc codes

In this letter, the stopping sets and stopping distance of finite geometry LDPC (FG-LDPC) codes are studied. It is known that FG-LDPC codes are majority-logic decodable and a lower bound on the minimum distance can be thus obtained. It is shown in this letter that this lower bound on the minimum distance of FG-LDPC codes is also a lower bound on the stopping distance of FG-LDPC codes, which implies that FG-LDPC codes have considerably large stopping distance. This may explain in one respect why some FG-LDPC codes perform well with iterative decoding in spite of having many cycles of length 4 in their Tanner graphs.

Improved weighted bit-flipping algorithm for decoding LDPC Codes

The paper presents an improvement of the weighted bit-flipping (WBF) decoding algorithm for LDPC codes proposed by Liu and Pados (2003). Simulation results show that the improved WBF algorithm provides about 0.25 dB coding gain over the Liu–Pados WBF algorithm. Also included in the paper is a simplified loop-detection algorithm, which is much easier to implement than the one proposed by Liu and Pados. The proposed WBF algorithm is particularly effective for decoding LDPC codes whose parity check matrices have large row/column weights such as finite geometry LDPC codes.

Projective-Plane Iteratively Decodable Block Codes for WDM High-Speed Long-Haul Transmission Systems

Low-density parity-check (LDPC) codes are excellent candidates for optical network applications due to their inherent low complexity of both encoders and decoders. A cyclic or quasi-cyclic form of finite geometry LDPC codes simplifies the encoding procedure. In addition, the complexity of an iterative decoder for such codes, namely the min-sum algorithm, is lower than the complexity of a turbo or Reed-Solomon decoder. In fact, simple hard-decoding algorithms such as the bit-flipping algorithm perform very well on codes from projective planes. In this paper, the authors consider LDPC codes from affine planes, projective planes, oval designs, and unitals. The bit-error-rate (BER) performance of these codes is significantly better than that of any other known foward-error correction techniques for optical communications. A coding gain of 9-10 dB at a BER of 10/sup -9/, depending on the code rate, demonstrated here is the best result reported so far. In order to assess the performance of the proposed coding schemes, a very realistic simulation model is used that takes into account in a natural way all major impairments in long-haul optical transmission such as amplified spontaneous emission noise, pulse distortion due to fiber nonlinearities, chromatic dispersion, crosstalk effects, and intersymbol interference. This approach gives a much better estimate of the code's performance than the commonly used additive white Gaussian noise channel model.