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IEEE/CIC International Conference on Communications in China (ICCC)

Construction of QC-LDPC Codes with Product-like Exponent Matrices Based on Rectangular Area

Jian Fang, Yuxiang Zhou, Min Zhu, Baoming Bai

Published Aug 10, 2025• 1 citation

Abstract

This paper proposes a novel algorithm for constructing girth-8 quasi-cyclic (QC) LDPC codes with product-like (PL) exponent matrices, termed PL-QC-LDPC codes. To simplify the analysis of cycles in the Tanner graph of PL-QC-LDPC codes, we introduce the rectangular area (RA) and present an efficient semi-search RA algorithm to determine the minimum size of circulant permutation matrices (CPMs) in the exponent matrix. Moreover, search depth is employed to balance the trade-off between search complexity and optimization of the minimum CPM-size. Numerical results demonstrate that our proposed method, combined with masking method, can achieve smaller CPM-size and lower error floor than the greatest common divisor (GCD) method.

DOI: 10.1109/icccworkshops67136.2025.11148120

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Recursive construction of

(J,L)

QC LDPC codes with girth 6

‎In this paper‎, ‎a recursive algorithm is presented to generate some exponent matrices which correspond to Tanner graphs with girth at least 6‎. ‎For a

J times L

exponent matrix

E

‎, ‎the lower bound

Q(E)

is obtained explicitly such that

(J,L)

QC LDPC codes with girth at least 6 exist for any circulant permutation matrix (CPM) size

m geq Q(E)

‎. ‎The results show that the exponent matrices constructed with our recursive algorithm have smaller lower-bound than the ones proposed recently with girth 6‎.

This article presents a simple, less computational complexity method for constructing exponent matrix (3, K) having girth at least 8 of quasi-cyclic low-density parity-check (QC-LDPC) codes based on subtraction method. The construction of code deals with the generation of exponent matrix by three formulas. This method is flexible for any block-column length K. The simulations are shown in comparison with some existing appreciable work. The codes with girth 8 are constructed with circulant permutation matrix (CPM) size P ≥ max{a2, r, a2, r,….a2, k} + 1.

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.

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