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IEEE Communications Letters

On the Construction of QC-LDPC Codes Based on Integer Sequence With Low Error Floor

Xiongfei Tao, Xin Chen, Bifang Wang

Published Jun 30, 2022• 5 citations
Publisher

Abstract

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.

DOI
10.1109/lcomm.2022.3187435

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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{a}a,b{b}b) witha≤10{a} \leq10a≤10andb≤{b} \leq b≤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.
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.
In this correspondence we present a special class of quasi-cyclic low-density parity-check (QC-LDPC) codes, called block-type LDPC (B-LDPC) codes, which have an efficient encoding algorithm due to the simple structure of their parity-check matrices. Since the parity-check matrix of a QC-LDPC code consists of circulant permutation matrices or the zero matrix, the required memory for storing it can be significantly reduced, as compared with randomly constructed LDPC codes. We show that the girth of a QC-LDPC code is upper-bounded by a certain number which is determined by the positions of circulant permutation matrices. The B-LDPC codes are constructed as irregular QC-LDPC codes with parity-check matrices of an almost lower triangular form so that they have an efficient encoding algorithm, good noise threshold, and low error floor. Their encoding complexity is linearly scaled regardless of the size of circulant permutation matrices.
Quasi-cyclic (QC) Low-density paritycheck (LDPC) codes are constructed from combination of weight-0 (null matrix) and Weight-2 (W2) Circulant matrix (CM), which can be seen as a special case of the general type-II QC LDPC codes. The shift matrix of the codes is built on the basis of one integer sequence, called perfect Cyclic difference set (CDS), which guarantees the girth of the code at least six. Simulation results show that the codes can perform well in comparison with a variety of other LDPC codes. They have excellent error floor and decoding convergence characteristics.
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.
This paper presents a novel regular Quasi-Cyclic (QC) Low Density Parity Check (LDPC) codes with column-weight three and girth at least eight. These are designed on the basis of combinatorial design in which subsets applied for the construction of circulant matrices are determined by a particular subset. Considering the non-existence of cycles four and six in the structure of the parity check matrix, a bound for their minimum weight is proposed. The simulations conducted confirm that without applying a masking technique, the newly implemented codes have a performance similar to or better than other well-known codes. This is evident in the waterfall region, while their error floor at very low Bit Error Rate (BER) is expected.
For any row weight L , a novel class of (8, L ) quasi‐cyclic low‐density parity‐check (QC‐LDPC) codes is deterministically constructed with girth eight. Compared with a similar construction, the proposed (7, L ) and (8, L ) QC‐LDPC codes exist for any circulant permutation matrix of size P ≥ ( L − 1)( L 2 + 3 L /4 + 3/4) + 1 and P ≥ ( L − 1)( L 2 + 3 L /4 + 7/4) +1, which remarkably improve the existing bounds of ( L − 1)( L 2 + L ) +1 and ( L − 1)( L 2 + L + 1) +1, respectively. Moreover, for any column weight J , any row weight L and any common difference, a construction for ( J , L ) QC‐LDPC codes with girth eight is also proposed. This construction has high flexibility with respect to the design of code rate and code length. Simulation results show that the novel codes with moderate lengths perform well in the additive white Gaussian noise channel.