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IEEE Communications Letters

On the Class of High-Rate QC-LDPC Codes With Girth 8 From Sequences Satisfied in GCD Condition

Marjan Majdzade, Mohammad Gholami

Published Mar 24, 2020• 11 citations
Publisher

Abstract

Recently, a class of (J, L) quasi-cyclic (QC) low-density parity-check (LDPC) codes with girth eight is constructed based on the greatest-common-divisor (GCD) condition. For L = 3, an equivalence between the sequences generated by a greedy algorithm satisfying in GCD condition and a known class of integer sequences, called Stanley sequences has been proposed as an open problem. In this letter, we solve this problem in a more general case by introducing the class of 3-free sets as a generalization of Stanley sequences and showing an equivalence between 3-free sets and the sequences satisfied in GCD condition. Then, a new algorithm is proposed to find 3-free sets efficiently usually having larger size than the known methods, leading to column-weight 3 QC-LDPC codes with smaller lengths and better 8-cycle distributions. In addition, a new explicit formula is proposed to construct Stanley sequences which results in a class of girth-8 column-weight 3 QC-LDPC codes with high rates. The protograph QC-LDPC codes lifted from the constructed base matrices outperform progressive-edge-growth (PEG), randomlike and some recent QC-LDPC codes with the same girth.

DOI
10.1109/lcomm.2020.2983019

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Recently, a class of (J, L) quasi-cyclic (QC) low-density parity-check (LDPC) codes with girth eight is constructed based on the greatest-common-divisor (GCD) condition. For L = 3, an equivalence between the sequences generated by a greedy algorithm satisfying in GCD condition and a known class of integer sequences, called Stanley sequences has been proposed as an open problem. In this letter, we solve this problem in a more general case by introducing the class of 3-free sets as a generalization of Stanley sequences and showing an equivalence between 3-free sets and the sequences satisfied in GCD condition. Then, a new algorithm is proposed to find 3-free sets efficiently usually having larger size than the known methods, leading to column-weight 3 QC-LDPC codes with smaller lengths and better 8-cycle distributions. In addition, a new explicit formula is proposed to construct Stanley sequences which results in a class of girth-8 column-weight 3 QC-LDPC codes with high rates. The protograph QC-LDPC codes lifted from the constructed base matrices outperform progressive-edge-growth (PEG), randomlike and some recent QC-LDPC codes with the same girth.
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.
The greatest-common-divisor (GCD) method is a general framework employing a set of simple inequalities (called GCD constraint) to guarantee girth eight for a class of (J,L) quasi-cyclic (QC) low-density parity-check (LDPC) codes. However, an important problem, i.e., whether the GCD constraint is necessary for this class of codes to have girth eight, remains open. In this letter, the question is answered affirmatively, following which a novel algorithm aiming to find the shortest codes with girth eight in such a class is proposed. Besides, a close connection is established between the GCD method and the base expansion method, which are both applicable for any J and any L.
Quasi-cyclic low-density parity-check (QC-LDPC) codes are known to have algebraic structure, which suits hardware implementation. However, low girth makes their performances worse than random structured LDPC codes. Based on independent row-column mapping sequence, this work proposed a generalized method of constructing QC-LDPC codes with girth eight, which enlarges the class of girth eight QC-LDPC codes to a fairly general series. Simulation results show that the bit error rate of the constructed codes is close to that of progressive edge growth (PEG) based QC-LDPC codes under iterative decoding.
Small cycles in spatially coupled low-density paritycheck (SC-LDPC) codes have a detrimental effect on the error floor. Recently, an explicit simple method has been proposed to construct a class of 4-cycle free SC-LDPC codes by partitioning a spreading matrix into a number of component matrices. Using a quasi-cyclic (QC) lifting, we propose a component matrices construction method to obtain QC-SC-LDPC codes with a girth eight in this paper. The component matrices of the new method are completely defined by a spreading matrix and an exponent matrix. The spreading matrix is constructed from a finite sequence of integers, and the exponent matrix is constructed from the greatest common divisor (GCD) framework. The novel construction method demonstrates the capacity to entirely eliminate 6-cycles. Compared to the recently proposed explicit structural design, the proposed code achieves a girth of eight and exhibits better bit-error rate (BER) performance.
Multitype quasi-cyclic (QC) low-density parity-check (LDPC) codes are a class of protograph LDPC codes lifted cyclically from protographs with multiple edges, represented by two weight and slope matrices. For a given weight-matrix, an approach is proposed to find the maximum-achievable girthgmax⁡g_{\max }gmax​of the corresponding multitype QC-LDPC codes by some inevitable chains having less complexity than the existing methods. This advantage leads to some new patterns of the weight matrices such that the corresponding codes have some improvements in terms of the maximum-achievable girths or the minimum-distance upper-bounds. In continue, for a given weight-matrix with maximum-achievable girthgmax⁡g_{\max }gmax​, some slope-matrices are constructed by a depth-first search algorithm for which the corresponding multitype QC-LDPC codes with even girthggg,g≤gmax⁡g\le g_{\max }g≤gmax​, have smaller lengths, higher rates, or larger minimum-distances than the state-of-the-art achievements. Simulation results show that the constructed codes have some error-rate advantages than PEG, random-like, CCSDS, and 802.11n/ac IEEE standard LDPC codes.
Multi-type quasi-cyclic (QC) low-density parity-check (LDPC) codes can be considered as multiple-edge protograph QC-LDPC codes having some advantages in the minimum Hamming distance bound over single-edge protograph codes or type-I QC-LDPC codes when the base matrices have the same size. In this paper, we investigate a class of multi-type QC-LDPC codes whose parity-check matrices contain just one blockrow of circulants and we obtain the generator matrix of such codes in general form. Using the permutation arrays and defining injection arrays, we present a new approach to construct a class of high-rate type-I QC-LDPC codes with girth 6 from the constructed 4-cycle free multi-type QC-LDPC codes. In continue, for 2 ≤ w ≤ 6, some type-w QC-LDPC codes with girth 6 are constructed explicitly such that the constructed codes are flexible in terms of rate and length. To the best of our knowledge, for w = 5,6, this is the first paper which deals with the explicit construction of type-w QC-LDPC codes with girth 6 and high rates. Moreover, for w = 3,4, the constructed type-w QC-LDPC codes have better (6,8)-cycle multiplicities than the codes with minimum achievable length recently constructed by cyclic difference families (CDFs). Simulation results show that the binary and non-binary constructed codes outperform the constituent underlying QC-LDPC codes.