Family Of Low-density Parity-check Codes Research Articles

New low-density-parity-check decoding approach based on the hard and soft decisions algorithms

<span lang="EN-US">It is proved that hard decision algorithms are more appropriate than a soft decision for low-density parity-check (LDPC) decoding since they are less complex at the decoding level. On the other hand, it is notable that the soft decision algorithm outperforms the hard decision one in terms of the bit error rate (BER) gap. In order to minimize the BER and the gap between these two families of LDPC codes, a new LDPC decoding algorithm is suggested in this paper, which is based on both the normalized min-sum (NMS) and modified-weighted bit-flipping (MWBF). The proposed algorithm is named normalized min sum- modified weighted bit flipping (NMSMWBF). The MWBF is executed after the NMS algorithm. The simulations show that our algorithm outperforms the NMS in terms of BER at 10-8 over the additive white gaussian noise (AWGN) channel by 0.25 dB. Furthermore, the proposed NMSMWBF and the NMS are both at the same level of decoding difficulty.</span>

Minimum distances of three families of low-density parity-check codes based on finite geometries

Three families of low-density parity-check (LDPC) codes are constructed based on the totally isotropic subspaces of symplectic, unitary, and orthogonal spaces over finite fields, respectively. The minimum distances of the three families of LDPC codes in some special cases are settled.

On the Smallest Absorbing Sets of LDPC Codes From Finite Planes

Absorbing sets, a class of combinatorial structures of the Tanner graph representation of a low-density parity-check (LDPC) code, are known to influence the performance of the code under message passing iterative decoding. In this paper, we study the smallest absorbing sets of LDPC codes constructed from projective planes and Euclidean planes. The lower bounds on the parameters of smallest absorbing sets given by Dolecek are proven to be tight for these two families of LDPC codes. We also analyze the combinatorial properties of the smallest absorbing sets and give conditions necessary and sufficient for a set of bit nodes in the Tanner graph to be a smallest absorbing set. For LDPC codes from projective planes, we further give a condition necessary and sufficient for a smallest absorbing set to be a fully absorbing set. In addition, we show that these smallest absorbing sets are asymptotically not stable, which may explain to some extent the good performance as well as the low error floor expectation of these two families of LDPC codes.

Girth-10 LDPC Codes Based on 3-D Cyclic Lattices

In this paper, we propose a new method based on combinatorial designs for constructing high-girth low-density parity-check (LDPC) codes. We use a 3-D lattice to generate balanced incomplete block designs based on planes and lines in the lattice. This gives families of regular LDPC codes with girths of at least 6, 8, and 10, whose parity-check matrices are all block circulant. The main advantage of this construction is that the algebraic structure leads to efficient encoders and decoders. Based on the block-circulant structure of a parity-check matrix, we present an efficient encoder that can be parallelized to improve the speed of encoding. The simulation results show that these families of LDPC codes perform very well on additive-white-Gaussian-noise channels (roughly 0.45 dB from the channel capacity) and Rayleigh fading channels (roughly 0.51 dB from the channel capacity).

LP Decoding Corrects a Constant Fraction of Errors

We show that for low-density parity-check (LDPC) codes whose Tanner graphs have sufficient expansion, the linear programming (LP) decoder of Feldman, Karger, and Wainwright can correct a constant fraction of errors. A random graph will have sufficient expansion with high probability, and recent work shows that such graphs can be constructed efficiently. A key element of our method is the use of a dual witness: a zero-valued dual solution to the decoding linear program whose existence proves decoding success. We show that as long as no more than a certain constant fraction of the bits are flipped by the channel, we can find a dual witness. This new method can be used for proving bounds on the performance of any LP decoder, even in a probabilistic setting. Our result implies that the word error rate of the LP decoder decreases exponentially in the code length under the binary-symmetric channel (BSC). This is the first such error bound for LDPC codes using an analysis based on "pseudocodewords." Recent work by Koetter and Vontobel shows that LP decoding and min-sum decoding of LDPC codes are closely related by the "graph cover" structure of their pseudocodewords; in their terminology, our result implies that that there exist families of LDPC codes where the minimum BSC pseudoweight grows linearly in the block length

Iteratively decodable codes from orthogonal arrays for optical communication systems

A novel family of low-density parity-check codes is proposed based on orthogonal arrays. Codes from this family have high code rate, girth of at least six, large minimum distance, and significantly outperform the error correction schemes based on turbo product codes proposed for optical communication systems.

Construction of well-structured quasi-cyclic low-density parity check codes

Two classes of well-structured binary low-density parity check codes (LDPC) are described. The first class is based on a branch of combinatorial mathematics, known as the balanced incomplete block design (BIBD). Construction of three types of codes derived from BIBD designs is illustrated in addition to a family of LDPC codes constructed by decomposition of incidence matrices of the proposed BIBD designs. The decomposition technique reduces the density of the parity check matrix and hence reduces the number of short cycles, which generally lead to better performing LDPC codes. The second class of well-structured LDPC codes, Vandermonde or array LDPC codes, are defined by a small number of parameters and cover a large set of code lengths and rates with different column weights. The presented LDPC codes are quasi-cyclic with no cycles of length four; hence simple encoding while maintaining good performance is achieved. Furthermore, the codes are compared with known random LDPC codes, in order to assess their relative achievable performance/complexity trade-offs. It is shown that well-structured LDPC codes perform very similar to the known random LDPC codes.

MacNeish–Mann Theorem Based Iteratively Decodable Codes for Optical Communication Systems

A novel family of low-density parity-check codes is proposed based on MacNeish-Mann theorem for construction of mutually orthogonal Latin squares. Codes from this family have high code rate, girth at least six, large minimum distance, and significantly outperform conventional forward error correction schemes based on Reed-Solomon (RS) and turbo codes.

High Code Rate Low-Density Parity-Check Codes for Optical Communication Systems

A novel family of low-density parity-check codes based on symmetrically repeated difference systems is proposed. Codes from this family have very high code rate, girth at least six, large minimum distance, and significantly outperform conventional Reed-Solomon codes.