Keke Liu

Algebraic Quasi-Cyclic LDPC Codes: Construction, Low Error-Floor, Large Girth and a Reduced-Complexity Decoding Scheme

This paper presents a simple and very flexible method for constructing quasi-cyclic (QC) low density paritycheck (LDPC) codes based on finite fields. The code construction is based on two arbitrary subsets of elements from a given field. Some well known constructions of QC-LDPC codes based on finite fields and combinatorial designs are special cases of the proposed construction. The proposed construction in conjunction with a technique, known as masking, results in codes whose Tanner graphs have girth 8 or larger. Experimental results show that codes constructed using the proposed construction perform well and have low error-floors. Also presented in the paper is a reduced-complexity iterative decoding scheme for QC-LDPC codes based on the section-wise cyclic structure of their parity-check matrices. The proposed decoding scheme is an improvement of an earlier proposed reduced-complexity iterative decoding scheme.

A Matrix-Theoretic Approach to the Construction of Non-Binary Quasi-Cyclic LDPC Codes

This paper presents two simple and very flexible methods for constructing non-binary (NB) quasi-cyclic (QC) LDPC codes. The proposed construction methods have several known ingredients including base array, masking, binary to nonbinary replacement, and matrix-dispersion. By proper choice and combination of these ingredients, NB-QC-LDPC codes with excellent performance can be constructed. The constructed codes can be decoded with a reduced-complexity iterative decoding scheme which significantly reduces the hardware implementation complexity.

Quasi-cyclic LDPC codes: Construction and rank analysis of their parity-check matrices

A construction of binary and non-binary quasi-cyclic (QC)-LDPC codes based on partitions of finite fields of characteristic 2 is proposed. The construction is carried out in the Fourier transform domain. The parity-check matrices of these QC-LDPC codes are arrays of circulant permutation matrices. The ranks of these arrays are analyzed and combinatorial expressions are derived. Example codes are given and simulations show that they perform well over the AWGN channel decoded with message-passing decoding algorithms.

Decoding of quasi-cyclic LDPC codes with section-wise cyclic structure

Presented in this paper is a reduced-complexity iterative decoding scheme for quasi-cyclic (QC) LDPC codes. This decoding scheme is devised based on the section-wise cyclic structure of the parity-check matrix of a QC-LDPC code. Using this decoding scheme, the hardware implementation complexity of a QC-LDPC decoder can be significantly reduced without performance degradation. A high-rate QC-LDPC code that can achieve a very low error-rate without a visible error-floor is used to demonstrate the effectiveness of the proposed decoding scheme. Also presented in this paper are two other high-rate QC-LDPC codes and a method for constructing rate -1/2 QC-LDPC codes whose Tanner graphs have girth 8. All the codes constructed perform well with low error-floor using the proposed decoding scheme.

Finite-Length Algebraic Spatially-Coupled Quasi-Cyclic LDPC Codes

The replicate-and-mask (R&M) construction of finite-length spatially-coupled (SC) LDPC codes is proposed in this paper. The proposed R&M construction generalizes the conventional matrix unwrapping construction and contains it as a special case. The R&M construction of a class of algebraic spatially coupled (SC) quasi-cyclic (QC) LDPC codes over arbitrary finite fields is demonstrated. The girth, rank, and time-varying periodicity of the proposed R&M SC QC LDPC codes are analyzed. The error rate performance of finite-length nonbinary algebraic SC QC LDPC codes is investigated with window decoding. Compared to the conventional unwrapping construction, it is found through numerical simulations that the R&M construction resulted in SC QC LDPC codes with better block error rate performance and lower error floors. With a flooding schedule decoder, it is shown that the proposed R&M algebraic SC QC LDPC codes have better error performance than the corresponding LDPC block codes and random SC codes. The R&M construction of irregular SC QC LDPC codes is demonstrated. It is shown that low-complexity regular puncturing schemes can be deployed on these codes to construct families of rate-compatible irregular SC QC LDPC codes with good performance.

Non-binary algebraic spatially-coupled quasi-cyclic LDPC codes.

This paper considers the algebraic construction and performance of non-binary spatially-coupled low density parity check (LDPC) codes. A replicate-and-mask approach is presented to construct finite-length algebraic quasi-cyclic (QC) spatially-coupled (SC) LDPC codes. Numerical results show the superiority of non-binary algebraic SC QC LDPC codes over the corresponding random non-binary (block and SC) LDPC codes. In this paper, it is demonstrated that the threshold saturation phenomenon, previously demonstrated for binary SC LDPC codes, also holds for non-binary SC LDPC codes over the binary-input AWGN channel with BPSK modulation.

Quasi-cyclic LDPC codes on two arbitrary sets of a finite field

This paper presents a simple and flexible method for constructing QC-LDPC codes based on two arbitrary sets of a finite field. Based on this method, a high-rate, high-performance and very low error-floor QC-LDPC code is first constructed and then a class of rate-1/2 QC-LDPC codes whose Tanner graphs have girth 8 or larger is presented. Also presented is a reduced-complexity iterative decoding algorithm for QC-LDPC codes.

Low-density arrays of circulant matrices: Rank and row-redundancy, and QC-LDPC codes

This paper is concerned with general analysis on the rank and row-redundancy of an array of circulants whose null space defines a QC-LDPC code. Based on the Fourier transform and the properties of conjugacy classes and Hadamard products of matrices, tight bounds on rank and row-redundancy are derived, which make it possible to consider row-redundancy in constructions of QC-LDPC codes to achieve better performance. Moreover, a new construction of QC-LDPC codes from random partitions of finite fields, which has flexible code dimensions and is abundant in row-redundancy, is presented and analyzed.

Construction of non-binary LDPC codes: A matrix-theoretic approach

This paper presents a matrix-theoretic approach to the construction of non-binary LDPC codes. Two algebraic methods for constructing non-binary LDPC codes are presented. The proposed construction methods have several ingredients including base matrix, matrix-dispersion, masking and replacement. By proper choice and combination of these ingredients, non-binary LDPC codes with good performance can be constructed.

A merry-go-round decoding scheme for non-binary quasi-cyclic LDPC codes

This paper presents a reduced-complexity iterative scheme and an algorithm for decoding non-binary quasi-cyclic (QC) LDPC codes of a specific type. The proposed decoding scheme and the algorithm together significantly reduce the hardware implementation complexity of a decoder with no performance degradation. Also presented in the paper is a simple method for constructing a class of non-binary QC-LDPC codes.