Column-Weight Three QC LDPC Codes with Girth 20

DOI: 10.1049/cp:20061337 ISBN: 0 86341 644 6 Location: Hangzhou, China Conference date: 6-9 Nov. 2006 Format: PDF The BER performance of presented quasi-cyclic (QC) low-density parity-check (LDPC) codes is not good as that of randomly constructed LDPC codes, while the lack of structure of randomly constructed LDPC codes implies serious disadvantages in terms of storing and accessing a large parity-check matrix. To solve the problems, this paper proposes a design of good quasi-cyclic (QC) low-density parity-check (LDPC) codes, the obtained algebraically structured codes have large minimum distances and good BER performance. The proposed design is based on index matrices, which determines the shifts of the circulant matrices of the sparse parity-check matrices. Compared with the presented design of QC LDPC codes, the designed QC LDPC codes are free from girth 4, sometimes free from girth 6, and have much larger minimum distances. Based on the algebraic code structure, the conditions of the girth and minimum distance of the codes are found. The BER performance of the designed QC LDPC block codes compares with that of randomly constructed LDPC codes for any block lengths. Better BER performance is obtained by increasing the circulant size of the base QC code. Simulations verify the construction of the QC LDPC codes to be valid. (4 pages) Inspec keywords: AWGN channels; error statistics; channel coding; algebraic codes; parity check codes; sparse matrices Subjects: Algebra; Other topics in statistics; Codes

In this paper we propose a method of constructing quasi-cyclic low-density parity-check (QC-LDPC) codes of large length by combining QC-LDPC codes of small length as their component codes, via the Chinese Remainder Theorem. The girth of the QC-LDPC codes obtained by the proposed method is always larger than or equal to that of each component code. By applying the method to array codes, we present a family of high-rate regular QC-LDPC codes with no 4-cycles. Simulation results show that they have almost the same performance as random regular LDPC codes.

Quasi-cyclic LDPC (QC-LDPC) codes have been accepted as the standard codes of 5G enhanced mobile broadband data channel. These standard codes are designed to support multiple lifting sizes and possess rate-compatible property, which can help adapt various information lengths and code rates well. In this paper, we propose an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">algebra-assisted</i> method for constructing QC-LDPC codes with such properties. We will first review the encoding mechanism and requirements of 5G LDPC codes, and present <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">cycle analysis</i> for such emerging codes. We then propose a metric, referred to as <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">weighted average number of cycles</i> (WANC), from the perspective of cycle structure for constructing the QC-LDPC codes that can support <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">multiple lifting sizes</i> . Based on the WANC metric and algebraic methods, we develop a simple and practical algorithm to construct this kind of QC-LDPC codes. We finally apply the proposed algorithm to construct the exponent matrices for cases of 5G LDPC codes and the standard LDPC codes of consultative committee for space data systems, respectively. Simulation results show that the proposed WANC metric and designed algorithm are feasible and effective, and thus can be utilized to design other similar QC-LDPC codes.

In this paper, we propose a simple mathematical structure to improve the BER (bit error rate) performance of quasi-cyclic low-density parity-check (QC-LDPC) codes. Since the proposed structure can be defined in-between those of random codes and structured codes, these proposed QC-LDPC codes not only perform as well as random-LDPC codes, but also preserve the advantage of structured LDPC codes which is easier to implement over coding hardware. Moreover, the proposed structure, constructed from balanced incomplete block design (BIBD), can reduce the density of parity-check matrix and hence reduce the number of short cycles, which generally lead to better performance. At the end, we show that the error floor performance of QC-LDPC codes can be improved by proposed hybrid structures and perform well over binary-input additive white Gaussian noise (BI-AWGN) channel.

In this letter, we consider two problems associated with quasi-cyclic low-density parity-check (QC-LDPC) codes. The first is how to extend the code length of a QC-LDPC code without reducing the girth. The second is how to design a QCLDPC code with a prescribed girth easily. We deal with these two problems by using a combining method of QC-LDPC codes via the Chinese Remainder Theorem (CRT). Codes constructed with our proposed method have flexible code lengths, flexible code rates and large girth. Simulation results show that they perform very well with the iterative decoding.

The generator matrix of quasi-cyclic low-density parity-check (QC-LDPC) codes is not, in general, in systematic form. Therefore, finding the message corresponding to a detected codeword at the receiver is not as straightforward as it is in the case of systematic codes. In this paper, we propose two methods for finding the message from the codeword in a QC-LDPC code. The first method is a general scheme which views the problem as a set of linear equations. This method is not realizable with pure digital circuits and may be implemented by having a processor core in the design. The second method is rather an alternative scheme which makes use of a particular structure in generator matrices of QC-LDPC codes in order to find the message from a non-systematic codeword. We examine a large class of non-systematic Latin squares QC-LDPC codes and one of the IEEE 802.16e standard QC-LDPC codes, and show that they have the required structure to be usable by the algorithm. The latter method is realizable in hardware and can be implemented with a digital circuit consisting of XOR gates.

Turbo codes are attractive compared with Low Density Parity Check (LDPC) codes for Forward Error Correction (FEC) applications mainly due to their superior performance, especially at low Signal-to-Noise Ratio (SNR) such as are common in Powerline channels. For example, IEEE 1901-FFT PHY used the Turbo coding scheme defined in the HomePlug AV standards. However, patent fees are usually required for each turbo-code enabled manufactured device. The objective of this paper is to examine whether unlicensed LDPC codes, with optimized choices of block lengths, could be a viable alternative for future Powerline Communications (PLC) applications. The paper shows that the performance of the LDPC codes can approximate that of the Turbo codes with higher block lengths, on channels with typical and realistic PLC characteristics. The paper also shows that the additional complexity associated with this increase in block length can be mitigated by the use of Quasi-Cyclic LDPC (QC-LDPC) codes.

Array based quasi-cyclic LDPC codes and their tight lower bounds on the lifting degree

Seeking good error correcting codes to improve the efficiency of continuous-variable quantum key distribution (CVQKD) reconciliation is a concerning issue. Due to the introduction of multidimensional reconciliation, the error correcting techniques in the classical binary-input additive white Gaussian noise channel are applicable to CVQKD. In this Letter, we apply the quasi-cyclic low-density parity-check (QC-LDPC) codes, which are specified in 5G protocol, to the reconciliation process. Simulation results show that the reconciliation efficiency can reach 92.6% when the code rate is 22/68 and the signal-to-noise ratio is 0.623. Such a new error correcting code points out a new direction for the development of CVQKD.

Recently, Low Density Parity-Check (LDPC) codes based on Affine Permutation Matrices (APM) drew lots of attention. Compared with the Quasi-Cyclic LDPC (QC-LDPC) codes, these kinds of codes have some advantages. APM-LDPC codes obtain lower cycle-distributions, minimum hamming distance and greater girth. This paper explains the importance of cyclic distribution by comparing APM-LDPC codes with QC-LDPC codes. Then a particular form of APM-LDPC codes is proposed and researched. The new codes can low down the cycle-distribution to larger extent. In the following research, an effective method, which constructs the new codes with fixed girth, is proposed. Simulations show that the construction method is reasonable and effective. The transmission performances are better than the traditional methods, as well. Finally, the implementation and verification are carried out on FPGA.

The layered scheme is famous for its efficiency and high throughput structure in decoding of LDPC codes, especially the quasi-cyclic LDPC codes, whose parity check matrices are made of cyclically-shifted identity matrices, which can be used as natural partitions of decoding layers. However, for many QC-LDPC codes, it leads to serious routing congestion if the cyclically shifted identity matrix size is taken directly as the layer partition and parallel factor in decoding. In this paper, a parity-check matrix reordering method is introduced, which can lower the decoding parallelism while keeping the layered decoding structure at the same time. The LDPC codes in DVB-S2/T2 standards are taken as an example to illustrate the proposed method, simulation and FPGA implementation shows that the method is effective in reducing the total decoder cost.

This letter gives a random construction for Low Density Parity Check (LDPC) codes, which uses an iterative algorithm to avoid short cycles in the Tanner graph. The construction method has great flexible choice in LDPC code’s parameters including codelength, code rate, the least girth of the graph, the weight of column and row in the parity check matrix. The method can be applied to the irregular LDPC codes and strict regular LDPC codes. Systemic codes have many applications in digital communication, so this letter proposes a construction of the generator matrix of systemic LDPC codes from the parity check matrix. Simulations show that the method performs well with iterative decoding.

International audience

The edge connections in factor graphs of quasi-cyclic (QC) LDPC codes, which affect their iterative decoding performance, are determined by their base matrices. In this paper, the effects of base matrices on the iterative decoding performance of QC-LDPC codes are investigated by using the recently proposed EXIT chart tool for protograph LDPC codes. Extensive experiments have been carried out and iterative decoding thresholds of QC-LDPC codes with the same degree distribution pair but base matrices having different structures and different dimensions have been obtained and compared. Experimental results show that 1) the base matrix structure affects not only the waterfall performance but also the error floor performance; base matrices having good waterfall performance may exhibit rather poor error floor performance. 2) by increasing the dimensions of base matrices for a given degree distribution, the threshold means are reduced while the ranges of threshold distributions are increased. 3) similar to the case of detailed represented LDPC codes, base matrices with sufficiently large dimensions may have thresholds surpassing the conventional ensemble threshold with the same degree distribution pair. Moreover, as a concrete example, an improved base matrix is designed for a rate-1/2 WiMax LDPC code.

Motivated by the works on the girth of Tanner (3,5), (3,7), (3,11), and (5,7) quasi-cyclic (QC) LDPC codes, we in this paper study the girth of Tanner (5,11) QC-LDPC codes. We first analyze the cycles of Tanner (5,11) QC-LDPC codes, and obtain the conditions for the existence of cycles of length less than 12 in Tanner (5,11) QC-LDPC codes of length 11p where p is a prime number and p =1 (mod 55). Notice that the condition is represented by the polynomial equations in a 55th root of unity of the prime field Fp. By checking the existence of solutions for these equations over Fp, the girths of Tanner (5,11) QC-LDPC codes are obtained.