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IEEE Information Theory Workshop (ITW)

Explicit Constructions of Girth-Eight QC-LDPC Codes: A Unified Framework Motivated by TDGS

Guohua Zhang, Yulin Hu, Defeng Ren, Yi Fang

Published Nov 24, 2024• 0 citations
Publisher

Abstract

A new search method named two-dimensional greedy search (TDGS) is proposed to progressively determine each element within an exponent matrix, so as to generate quasi-cyclic (QC) low-density parity-check (LDPC) codes without 4-cycles and 6-cycles. By applying four different two-dimensional scanning orders to the TDGS, a design framework is formulated to unify several existing and novel explicit constructions for short (3,$L$)-regular QC-LDPC codes without 4-cycles and 6-cycles. Through equivalence analysis between the exponent matrices greedily found by the TDGS and directly defined by explicit constructions, novel girth properties of (and connections between) these existing explicit constructions are also revealed.

DOI
10.1109/itw61385.2024.10806926

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Dianzi xuebao•Jan 1, 2012•3 citations
A new search method named two-dimensional greedy search (TDGS) is proposed to progressively determine each element within an exponent matrix, so as to generate quasi-cyclic (QC) low-density parity-check (LDPC) codes without 4-cycles and 6-cycles. By applying four different two-dimensional scanning orders to the TDGS, a design framework is formulated to unify several existing and novel explicit constructions for short (3,LLL)-regular QC-LDPC codes without 4-cycles and 6-cycles. Through equivalence analysis between the exponent matrices greedily found by the TDGS and directly defined by explicit constructions, novel girth properties of (and connections between) these existing explicit constructions are also revealed.
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.
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.
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.
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.
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.
Enabling the parity-check matrices to have a large girth is an efficient method to improve the decoding performance for many binary and non-binary QC-LDPC codes.A new explicit framework to construct QC-LDPC codes with girth at least eight is proposed,without any computer search procedures.The QC-LDPC codes constructed by the framework not only have a girth at least eight,but also possess an advantage of consecutive cyclic permutation matrix(CPM) sizes.The framework includes two steps:first,a parity-check matrix is explicitly constructed with a girth at least eight under an infinite CPM size;second,the tight lower bound of consecutive CPM sizes is precisely calculated directly from the corresponding shift matrix of the parity-check matrix,by employing a new girth property discovered by this paper.