Quasi-cyclic low-density parity-check (QC-LDPC) codes have efficient hardware implementation and excellent correcting performance. However, the existence of trapping sets can affect the decoding performance. In this letter, a novel class of QC-LDPC codes based on integer sequence with column weight 3 and girth at least 8 is proposed. If the numbers in the sequence are different, the QC-LDPC codes free of elementary trapping sets ( ${a}$ , ${b}$ ) with ${a} \leq10$ and${b} \leq $3 can be constructed. The row weight of the parity check matrix can be arbitrary and we give a lower bound of the circulant permutation matrix (CPM) size. Simulation results show that the generated codes have good performance with low error floor over AWGN channels.Quasi-cyclic low-density parity-check (QC-LDPC) codes have efficient hardware implementation and excellent correcting performance. However, the existence of trapping sets can affect the decoding performance. In this letter, a novel class of QC-LDPC codes based on integer sequence with column weight 3 and girth at least 8 is proposed. If the numbers in the sequence are different, the QC-LDPC codes free of elementary trapping sets (${a}$,${b}$) with${a} \leq10$and${b} \leq $3 can be constructed. The row weight of the parity check matrix can be arbitrary and we give a lower bound of the circulant permutation matrix (CPM) size. Simulation results show that the generated codes have good performance with low error floor over AWGN channels.

IEEE Communications LettersJun 30, 20225 citations

Design of QC-LDPC Codes at Finite Lengths

Hyunki Kim, Eugene Ha, Jeongseok Ha

Sep 9, 20260 citations

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

Marjan Majdzade, Mohammad Gholami

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.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.

IEEE Communications LettersMar 24, 202011 citations

Constructions of LDPC Codes Based on Finite Fields

W.E. Ryan, Shu Lin

Cambridge University Press eBooksSep 17, 20090 citations

Explicit constructions for type-1 QC-LDPC codes with girth at least ten

Juhua Wang, Guohua Zhang, Quan Zhou +2 more

From Sidon sequence over ZP, two classes of type-1 (J = 3;L) QC-LDPC codes are explicitly proposed with block length L2P and with girth at least ten. For the first method, any Sidon sequence over ZP(P odd) with cardinality no smaller than L (L odd) corresponds to a class of type-1 (3, L) QC-LDPC codes with girth at least ten. For the second one, any Sidon sequence over ZP(P prime) with cardinality no smaller than L (L arbitrary) corresponds to a class of type-1 (3, L) QC-LDPC codes with girth at least ten. Compared with the method from 3-D lattices, which can also generate type-1 QC-LDPC codes with girth at least ten, the main advantage of the new methods is that the construction process is much simpler, and L is not necessarily a prime integer. Simulation results show that the new type-1 QC-LDPC codes outperform the girth-8 QC-LDPC codes constructed by Sidon sequence or the earliest sequence, and perform almost as well as Bocharova's shortest girth-12 QC-LDPC codes.From Sidon sequence over ZP, two classes of type-1 (J = 3;L) QC-LDPC codes are explicitly proposed with block length L2P and with girth at least ten. For the first method, any Sidon sequence over ZP(P odd) with cardinality no smaller than L (L odd) corresponds to a class of type-1 (3, L) QC-LDPC codes with girth at least ten. For the second one, any Sidon sequence over ZP(P prime) with cardinality no smaller than L (L arbitrary) corresponds to a class of type-1 (3, L) QC-LDPC codes with girth at least ten. Compared with the method from 3-D lattices, which can also generate type-1 QC-LDPC codes with girth at least ten, the main advantage of the new methods is that the construction process is much simpler, and L is not necessarily a prime integer. Simulation results show that the new type-1 QC-LDPC codes outperform the girth-8 QC-LDPC codes constructed by Sidon sequence or the earliest sequence, and perform almost as well as Bocharova's shortest girth-12 QC-LDPC codes.

IEEE Information Theory Workshop (ITW)Nov 1, 201410 citations

{a}

{b}

) with

{a} \leq10

and${b} \leq $3 can be constructed. The row weight of the parity check matrix can be arbitrary and we give a lower bound of the circulant permutation matrix (CPM) size. Simulation results show that the generated codes have good performance with low error floor over AWGN channels.