Finite Geometry LDPC Codes Research Articles

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.

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.

Cyclic and Quasi-Cyclic LDPC Codes on Constrained Parity-Check Matrices and Their Trapping Sets

This paper is concerned with construction and structural analysis of both cyclic and quasi-cyclic codes, particularly low-density parity-check (LDPC) codes. It consists of three parts. The first part shows that a cyclic code given by a parity-check matrix in circulant form can be decomposed into descendant cyclic and quasi-cyclic codes of various lengths and rates. Some fundamental structural properties of these descendant codes are developed, including the characterization of the roots of the generator polynomial of a cyclic descendant code. The second part of the paper shows that cyclic and quasi-cyclic descendant LDPC codes can be derived from cyclic finite-geometry LDPC codes using the results developed in the first part of the paper. This enlarges the repertoire of cyclic LDPC codes. The third part of the paper analyzes the trapping set structure of regular LDPC codes whose parity-check matrices satisfy a certain constraint on their rows and columns. Several classes of finite-geometry and finite-field cyclic and quasi-cyclic LDPC codes with large minimum distances are shown to have no harmful trapping sets of size smaller than their minimum distances. Consequently, their error-floor performances are dominated by their minimum distances.

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.

A differential binary message-passing LDPC decoder

In this paper, we propose a binary message-passing algorithm for decoding low-density parity-check (LDPC) codes. The algorithm substantially improves the performance of purely hard-decision iterative algorithms with a small increase in the memory requirements and the computational complexity. We associate a reliability value to each nonzero element of the code's parity-check matrix, and differentially modify this value in each iteration based on the sum of the extrinsic binary messages from the check nodes. For the tested random and finitegeometry LDPC codes, the proposed algorithm can perform as close as about 1 dB and 0.5 dB to belief propagation (BP) at the error rates of interest, respectively. This is while, unlike BP, the algorithm does not require the estimation of channel signal to noise ratio. Low memory and computational requirements and binary message-passing make the proposed algorithm attractive for high-speed low-power applications.

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.

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.

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.

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.

Parallel Weighted Bit-Flipping Decoding

A parallel weighted bit-flipping (PWBF) decoding algorithm for low-density parity-check (LDPC) codes is proposed. Compared to the best known serial weighted bit-flipping decoding, the PWBF decoding converges significantly faster but with little performance penalty. For decoding of finite-geometry LDPC codes, we demonstrate through examples that the proposed PWBF decoding converges in about 5 iterations with performance very close to that of the standard belief-propagation decoding.

An Improved Decoding Algorithm for Finite-Geometry LDPC Codes

In this letter, we propose a new modification to the belief propagation (BP) decoding algorithm for Finite-Geometry low-density parity-check (LDPC) codes. The modification is based on introducing into the iterative process, which can break the oscillations of bit log-likelihood ratio (LLR) values. Simulations show that, with a given maximum iteration, the feedback BP (FBP) algorithm can achieve better performance than the conventional belief propagation algorithm.

Tree-Based Construction of LDPC Codes Having Good Pseudocodeword Weights

We present a tree-based construction of low-density parity-check (LDPC) codes that have minimum pseudocodeword weight equal to or almost equal to the minimum distance, and perform well with iterative decoding. The construction involves enumerating a d-regular tree for a fixed number of layers and employing a connection algorithm based on permutations or mutually orthogonal Latin squares to close the tree. Methods are presented for degrees d=ps and d=ps+1, for p a prime. One class corresponds to the well-known finite-geometry and finite generalized quadrangle LDPC codes; the other codes presented are new. We also present some bounds on pseudocodeword weight for p-ary LDPC codes. Treating these codes as p-ary LDPC codes rather than binary LDPC codes improves their rates, minimum distances, and pseudocodeword weights, thereby giving a new importance to the finite-geometry LDPC codes where p>2

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.

A Decoding Algorithm for Finite-Geometry LDPC Codes

In this paper, we develop a new low-complexity algorithm to decode low-density parity-check (LDPC) codes. The developments are oriented specifically toward low-cost, yet effective, decoding of (high-rate) finite-geometry (FG) LDPC codes. The decoding procedure updates iteratively the hard-decision received vector in search of a valid codeword in the vector space. Only one bit is changed in each iteration, and the bit-selection criterion combines the number of failed checks and the reliability of the received bits. Prior knowledge of the signal amplitude and noise power is not required. An optional mechanism to avoid infinite loops in the search is also proposed. Our studies show that the algorithm achieves an appealing tradeoff between performance and complexity for FG-LDPC codes.

A Decoding Algorithm for Finite-Geometry LDPC Codes

A Decoding Algorithm for Finite-Geometry LDPC Codes

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.

Low-density parity-check codes based on finite geometries: a rediscovery and new results

This paper presents a geometric approach to the construction of low-density parity-check (LDPC) codes. Four classes of LDPC codes are constructed based on the lines and points of Euclidean and projective geometries over finite fields. Codes of these four classes have good minimum distances and their Tanner (1981) graphs have girth 6. Finite-geometry LDPC codes can be decoded in various ways, ranging from low to high decoding complexity and from reasonably good to very good performance. They perform very well with iterative decoding. Furthermore, they can be put in either cyclic or quasi-cyclic form. Consequently, their encoding can be achieved in linear time and implemented with simple feedback shift registers. This advantage is not shared by other LDPC codes in general and is important in practice. Finite-geometry LDPC codes can be extended and shortened in various ways to obtain other good LDPC codes. Several techniques of extension and shortening are presented. Long extended finite-geometry LDPC codes have been constructed and they achieve a performance only a few tenths of a decibel away from the Shannon theoretical limit with iterative decoding.