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IEEE Transactions on Communications

Symmetrical Constructions for Regular Girth-8 QC-LDPC Codes

Alireza Tasdighi, Amir H. Banihashemi, Mohammad‐Reza Sadeghi

Published Jan 1, 2016• 66 citations
Publisher

Abstract

In this paper, we propose new constructions for regular girth-8 quasi-cyclic low-density parity-check (QC-LDPC) codes based on circulant permutation matrices (CPM). The constructions assume symmetries in the structure of the parity-check matrix and employ a greedy exhaustive search algorithm to find the permutation shifts of the CPMs. As a result of symmetries, the new codes have a more compact representation compared with their counterparts. In majority of cases, also, they achieve the girth 8 at a shorter block length for the same degree distribution (code rate). Deterministic (explicit) constructions are also presented to expand the proposed parity-check matrices to larger block lengths and higher rates. The proposed long high-rate codes are often substantially shorter than regular girth-8 QC-LDPC codes of similar rate in the literature. Simulation results demonstrate that the proposed symmetric codes have competitive performance in comparison with similar existing QC-LDPC codes that lack symmetry.

DOI
10.1109/tcomm.2016.2617335

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In this paper, we propose new constructions for regular girth-8 quasi-cyclic low-density parity-check (QC-LDPC) codes based on circulant permutation matrices (CPM). The constructions assume symmetries in the structure of the parity-check matrix and employ a greedy exhaustive search algorithm to find the permutation shifts of the CPMs. As a result of symmetries, the new codes have a more compact representation compared with their counterparts. In majority of cases, also, they achieve the girth 8 at a shorter block length for the same degree distribution (code rate). Deterministic (explicit) constructions are also presented to expand the proposed parity-check matrices to larger block lengths and higher rates. The proposed long high-rate codes are often substantially shorter than regular girth-8 QC-LDPC codes of similar rate in the literature. Simulation results demonstrate that the proposed symmetric codes have competitive performance in comparison with similar existing QC-LDPC codes that lack symmetry.
In order to construct girth-8 quasi-cyclic (QC) low-density parity-check (LDPC) codes with low encoding complexity, this paper introduces structural shaping techniques to generate QC-LDPC codes with two different structures. These two structures are QC-LDPC codes constructed with vertical symmetric structure and reverse structure without 4-cycles and 6-cycles. Novel constructions assume that the structures of the parity check matrix (PCM) are symmetric and reverse, respectively, and the greedy exhaustive search algorithm is used to find the exponent matrix with the smallest possible size P of the circulant permutation matrix (CPM). Due to its symmetrical and reverse structural characteristics, the representation of the new codes is more compact than other codes. The simulation results show that the proposed codes have certain competitiveness compared to existing QC-LDPC codes.
By classifying cycle types within the parity-check matrix of a quasi-cyclic(QC) low-density parity-check(LDPC) code,a method without repetition or omission is proposed to detect all the cycles with lengths smaller than twelve.Based on this cycle detection method,a new approach is presented to construct large girth QC-LDPC codes by using some heuristic strategies.The approach includes three steps.Firstly,under the infinite cyclic permutation matrix(CPM) size condition,two integer sequences satisfying the girth constraint are found using a heuristic strategy.Secondly,according to the designed code rate,several integer pairs are extracted from the two sequences so as to form a shift matrix.Finally,by employing the recently proposed theoretical lower bound by Zhang et.al,a range of consecutive CPM sizes in which the girth never decrease is accurately computed from the shift matrix.Compared with Liu-Han's method,the new approach possesses two advantages in the sense that:(1) Not only girth-10+ QC-LDPC codes but also girth-12 ones can be constructed to have consecutive CPM sizes;(2) girth-10+ and girth-12 QC-LDPC codes can be generated with very high code rates,due to the new cycle detection method which significantly accelerates the sequence search procedure.
To address the challenge of constructing short girth-8 quasi-cyclic (QC) low-density parity-check (LDPC) codes, a novel construction framework based on vertical symmetry (VS) is proposed. Basic properties of the VS structure are presented. With the aid of these properties, existing explicit constructions for column weights from three to five which can be transformed into the VS structure are sorted out. Then two novel explicit constructions with the VS structure which guarantee short codes are presented for column weights of three and six. Moreover, an efficient search-based method is also proposed to find short codes with the VS structure. Compared with the state-of-the-art benchmarks, both the explicit constructions and the search-based method presented in this paper can provide shorter codes for most cases. Simulation results show that the new shorter codes can perform almost the same as or better than the longer existing counterparts. Thus, the new shorter codes can fit better with the low-latency requirement for modern communication systems.
To address the challenge of constructing short girth-8 quasi-cyclic (QC) low-density parity-check (LDPC) codes, a novel construction framework based on vertical symmetry (VS) is proposed. Basic properties of the VS structure are presented. With the aid of these properties, existing explicit constructions for column weights from three to five which can be transformed into the VS structure are sorted out. Then two novel explicit constructions with the VS structure which guarantee short codes are presented for column weights of three and six. Moreover, an efficient search-based method is also proposed to find short codes with the VS structure. Compared with the state-of-the-art benchmarks, both the explicit constructions and the search-based method presented in this paper can provide shorter codes for most cases. Simulation results show that the new shorter codes can perform almost the same as or better than the longer existing counterparts. Thus, the new shorter codes can fit better with the low-latency requirement for modern communication systems.
Short cycles in the Tanner graph of a given LDPC code has negative effect on the performance of iterative decoding. To improve the performance, we presented an approach for constructing Quasi-Cyclic Low-Density Parity-Check (QC-LDPC) codes whose associated Tanner graph has large girth and a small amount of short cycles. Each column block of the parity check matrix was designed by adding circulant permutation sub-matrices one after another, provided the current parity check matrix has minimal number of short cycles. The condition for the proposed QC-LDPC codes to have girth at least 6 was also analyzed. Regular and irregular QC-LDPC codes with various lengths and rates can be constructed. Simulations show that the proposed QC-LDPC codes have less short cycles and better performance than existing progressive edge growth based QC-LDPC codes.