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IEEE International Conference on Communication Technology (ICCT)

New Construction of QC-LDPC Codes Based on Full-Length Row-Multiplier

Ke Jia, Yu Qin, Mei Li

Published Oct 18, 2024• 0 citations
Publisher

Abstract

This paper proposes a new explicit construction method for quasi-cyclic (QC) low-density parity-check (LDPC) codes without 4-cycles and 6-cycles with large column weights. Two integer sequences entirely define the exponent matrix of the new method. The first sequence is an arithmetic sequence starting from 0 with a tolerance of 1, and the second sequence is a special sequence composed of integers that meet the constraint of the greatest common divisor (GCD). The existing explicit method can only provide a variety of row weight types with large circulant size. Compared with the recently proposed symmetric structure method based on computer search, the new explicit construction method has better decoding performance, extremely low description complexity, and does not require computer search.

DOI
10.1109/icct62411.2024.10946320

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This paper proposes a new explicit construction method for quasi-cyclic (QC) low-density parity-check (LDPC) codes without 4-cycles and 6-cycles with large column weights. Two integer sequences entirely define the exponent matrix of the new method. The first sequence is an arithmetic sequence starting from 0 with a tolerance of 1, and the second sequence is a special sequence composed of integers that meet the constraint of the greatest common divisor (GCD). The existing explicit method can only provide a variety of row weight types with large circulant size. Compared with the recently proposed symmetric structure method based on computer search, the new explicit construction method has better decoding performance, extremely low description complexity, and does not require computer search.
For any column weight J and any row weight L, a novel framework is proposed such that a girth-eight (J,L) quasi-cyclic low-density parity-check (QC-LDPC) code with any block length above a lower bound can be constructed via a simple inequality in terms of greatest common divisor (GCD). The main advantage is that the construction of a class of (J,L) girth-eight QC-LDPC codes is transformed into a rather simple task, searching for J integers satisfying the so-called GCD constraint for L. Combining the new method with masking matrices, a class of type-1 QC-LDPC codes is presented with girth at least eight. Simulation results show that the type-1 codes perform better than the random QC-LDPC codes and quadratic-congruence-based QC-LDPC codes for moderate block lengths and low code rates.
As a special case of a quasi-cyclic (QC) low-density parity-check (LDPC) code, a full-length row-multiplier (FLRM) QC-LDPC code is described by a compact exponent matrix based on two sequences of integers. The codes designed by a framework known as greatest-common-divisor (GCD) method, belong to a salient class of FLRM QC-LDPC codes, which can eliminate cycles of length up to six by carefully selecting a special sequence subject to a set of simple inequalities. However, the GCD method ensures the absence of these cycles only if circulant sizes are larger than a certain threshold. By combining the existing GCD method, novel sequences and a new analysis method (based on new lemmas of circulants and integers) for modulo equations, a group of novel FLRM QC-LDPC codes free of 4-cycles and 6-cycles are explicitly proposed for column weights from three to five in this paper, which possess circulant sizes much smaller than the forgoing threshold. Simulations show that the new FLRM QC-LDPC codes with shorter lengths perform almost the same as the existing FLRM QC-LDPC codes with longer lengths, and that the novel FLRM QC-LDPC codes noticeably outperform their counterparts with (nearly) identical lengths.
For a high-rate case, it is difficult to randomly construct good low-density parity-check (LDPC) codes of short and moderate lengths because their Tanner graphs are prone to have short cycles. Also, the existing high-rate quasi-cyclic (QC) LDPC codes can be constructed only for very restricted code parameters. In this paper, based on special classes of cyclic difference families, we propose a new construction method of high-rate regular QC LDPC codes having parity-check matrices consisting of a single row of circulants with column-weight 3 or 4. The proposed QC LDPC codes can be constructed for various code rates and lengths including the minimum achievable length for given column-weight and design rate under girth 6. It is observed that the parity-check matrices of the proposed QC LDPC codes have full rank for column-weight 3 and just one redundant row for column-weight 4. It is shown that the error correcting performance of the proposed QC LDPC codes of short and moderate lengths is almost the same as that of the existing ones through numerical analysis.
A novel class of quasi-cyclic (QC) low-density parity-check (LDPC) codes is explicitly proposed without 4-cycles and 6-cycles via the greatest-common-divisor (GCD) framework. From the new codes, a novel type of girth-eight QC-LDPC codes with a dual-diagonal (DD) structure is presented, in which the single undetermined entry in the associated exponent matrix can be determined by simple formulae. The proposed DD codes have similar or better performance compared with DD codes obtained from computer search. Moreover, a lower bound on circulant sizes which guarantees the existence of qualified undetermined entries is also provided.
Quasi-Cyclic Low-Density Parity-Check (QC-LDPC) codes have practical significance in the field of engineering, thus their construction algorithm is a hot issue in the field of LDPC codes. Based on existing construction methods, particularly the method based on Progressive Edge-Growth (PEG) algorithm, a new construction method of shift matrix is proposed. It can effectively avoid the copy of cycle structures, especially short cycles in base matrix after extension. Besides, it also relieves the time cost caused by random search, and greatly improves the length of cycles when compared with algorithms based on methods such as quadratic congruence and arithmetic sequence. Those algorithms can only eliminate short cycles with length 4, while our method can provide QC-LDPC codes without length 6 at least. Cycle analysis and simulation results show that the newly constructed QC-LDPC codes have good cycle structure and error correction performance.
Low-density Parity-check (LDPC) coding is a class of advanced channel coding techniques. To improve the performance of LDPC codes, a simple and effective method to construct QC-LDPC codes without short cycles (cycles of length four and six) is proposed. The approach is especially effective when the parity-check matrix of QC-LDPC codes is composed of circulant permutation matrices. Firstly, we analyze the shapes of the cycles in the exponent matrix, to get the required conditions of no four-cycles and six-cycles. Then we show how to construct the parity-check matrix without four-cycles and six-cycles by filling rows of the exponent matrix with arithmetic sequences. Simulation results show that the proposed codes have better performance than array codes. Though the performance of the proposed codes is not as well as random codes at higher Eb/N0, they can be applied to practical communications because of being encoded in linear time with shift registers.