Haiyang Liu

On the Number of Minimum Stopping Sets and Minimum Codewords of Array LDPC Codes

For an odd prime q and an integer m ≤ q, a binary mq × q² quasi-cyclic parity-check matrix H(m,q) can be constructed for a class of array LDPC code C(m,q). In this letter, the closed-form formula for the numbers of minimum stopping sets of H(m,q) and minimum codewords of C(m,q) are given for 2 ≤ m ≤ 3 and for (m,q)=(4,5) and (4,7).

Quasi-Cyclic Representation and Vector Representation of RS-LDPC Codes

RS-LDPC codes, constructed based on the codewords of Reed-Solomon (RS) codes with two information symbols, are an important class of LDPC codes. In this paper, we present two representations, namely, quasi-cyclic (QC) representation and vector representation, for RS-LDPC codes. Under the first representation, most part of the parity-check matrix of a full-length RS-LDPC code consists of circulant permutation matrices and zero matrices. As a result, the class of codes can enjoy the advantages in hardware implementation as QC-LDPC codes. In addition, the base matrix under the QC representation of an RS-LDPC code can be explicitly given such that the rank of its parity-check matrix can be analyzed combinatorially. Under the second representation, each permutation matrix in the parity-check matrix of an RS-LDPC code is defined by a nonbinary vector, whose entries are a permutation of entries in the field from which the RS code is constructed. Then, the “affine invariance” property is proved for full-length RS-LDPC codes, which can facilitate the structural analysis of the codes.

Novel modified min-sum decoding algorithm for low-density parity-check codes

The problem of improving the performance of min-sum decoding of low-density parity-check (LDPC) codes is considered in this paper. Based on min-sum algorithm, a novel modified min-sum decoding algorithm for LDPC codes is proposed. The proposed algorithm modifies the variable node message in the iteration process by averaging the new message and previous message if their signs are different. Compared with the standard min-sum algorithm, the modification is achieved with only a small increase in complexity, but significantly improves decoding performance for both regular and irregular LDPC codes. Simulation results show that the performance of our modified decoding algorithm is very close to that of the standard sum-product algorithm for moderate length LDPC codes.

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.

On the Decomposition Method for Linear Programming Decoding of LDPC Codes

Research on Wireless Sensor Networks (WSN) has evolved with potential applications in several domains. However, the building of WSN applications is hampered by the need of programming in low-level abstractions provided by sensor OS and of specific knowledge about each application domain and each sensor platform. We propose a MDA approach to develop WSN applications. This approach allows domain experts to directly contribute in the development of applications without needing low level knowledge on WSN platforms and, at the same time, it allows network experts to program WSN nodes to met application requirements without specific knowledge on the application domain. Our approach also promotes the reuse of the developed software artifacts, allowing an application model to be reused across different sensor platforms and a platform model to be reused for different applications.In this paper, we focus on solving the linear programming (LP) problem that arises in the decoding of low-density parity-check (LDPC) codes by means of the revised simplex method. In order to take advantage of the structure of the LP problem, we reformulate the dual LP and apply the idea of Dantzig-Wolfe (D-W) decomposition method to solve the problem. Each subproblem in the D-W decomposition method is an LP over a convex polyhedral cone. We define the convex polyhedral cone as local parity-check cone and discuss its special structures. Then, we enumerate its extreme rays and use these extreme rays to design an efficient method for the general LP decoding problem. The proposed method exhibits advantages in reducing both the storage and computational requirements.

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.

More on the Minimum Distance of Array LDPC Codes

Array low-density parity-check (LDPC) codes are a special class of quasi-cyclic LDPC codes constructed from a pair of integers (m, q), where q is an odd prime, and m ≤ q. In this paper, an improved upper bound on the minimum distance of this class of codes is given.

On the Minimum Distance of Full-Length RS-LDPC Codes

Let q be a power of 2 and y ≤ q an integer. Based on the codewords of [q, 2, q - 1] extended Reed-Solomon (RS) code over the finite field Fq, we can construct a (γ, q)-regular low-density parity-check (LDPC) code, called a full-length RS-LDPC code and denoted by C(γ, q). In this letter, the minimum distance of these codes is investigated. For any given q and y <; q, an upper bound on d(C(γ, q)), the minimum distance of C(γ, q), is provided. Furthermore, we determine the values of d(C(γ, q)) for y = 2, 3, and 4, and present the closed-form expressions on the numbers of minimum-weight codewords in C(γ, q) for γ = 2 and 3.

Multistep linear programming approaches for decoding low-density parity-check codes

Abstract The problem of improving the performance of linear programming (LP) decoding of low-density parity-check (LDPC) codes is considered in this paper. A multistep linear programming (MLP) algorithm was developed for decoding LDPC codes that includes a slight increase in computational complexity. The MLP decoder adaptively adds new constraints which are compatible with a selected check node to refine the results when an error is reported by the original LP decoder. The MLP decoder result is shown to have the maximum-likelihood (ML) certificate property. Simulations with moderate block length LDPC codes suggest that the MLP decoder gives better performance than both the original LP decoder and the conventional sum-product (SP) decoder.

On the Spark of Binary LDPC Measurement Matrices From Complete Protographs

The spark is an important property that provides recovery guarantees of a measurement matrix in compressed sensing. In this letter, we focus on a specific class of measurement matrices, i.e., binary low-density parity-check (LDPC) matrices from complete protographs. An upper bound on the spark of a binary LDPC measurement matrix from complete protograph is provided. In addition, the spark of array-based LDPC measurement matrices with certain column weights is analyzed.