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International Conference on Communications

Construction of protographs for large-girth structured LDPC convolutional codes

Junho Cho, Laurent Schmalen

Published Jun 1, 2015• 31 citations
Publisher

Abstract

In this paper, we present a method to construct girth-6 protographs that lead to the shortest constraint length in the convolutional structure. A stringent structural constraint is imposed on the protographs such that the decoder can be implemented efficiently. Then, given the structural constraint, it is shown that finding the aforementioned protographs is equivalent to solving a simple algebraic problem. Based on this mathematical formulation, girth-6 protographs are created without having to resort to a graph search. Using the girth-6 protographs, we derive good low-density parity-check (LDPC) convolutional codes by using periodic quasi-cyclic lifting. The performance of such constructed codes is compared with AR4JA-based LDPC convolutional codes.

DOI
10.1109/icc.2015.7249017

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In this paper, we present a method to construct girth-6 protographs that lead to the shortest constraint length in the convolutional structure. A stringent structural constraint is imposed on the protographs such that the decoder can be implemented efficiently. Then, given the structural constraint, it is shown that finding the aforementioned protographs is equivalent to solving a simple algebraic problem. Based on this mathematical formulation, girth-6 protographs are created without having to resort to a graph search. Using the girth-6 protographs, we derive good low-density parity-check (LDPC) convolutional codes by using periodic quasi-cyclic lifting. The performance of such constructed codes is compared with AR4JA-based LDPC convolutional codes.
In this paper, we deal with time-varing quasi-cyclic (QC) spatially-coupled (SC) low-density parity-check (LDPC) codes by lifting the coupled protographs. The construction method of girth-6 protographs for any code rate is proposed. We provide a theoretical lower bound on the syndrome former constraint length under condition of the protographs without 4-cycles in their Tanner graphs. Then we use periodic QC lifting method to search the QC-SC-LDPC matrices with larger girth based on the protographs. The minimum dimension size of QC-SC-LDPC codes is given to count the girth of QC-SC-LDPC codes with a semi- infinite binary-check matrix structure. The performance of such constructed codes with lower complexity and delay is superior to the LDPC block codes.
We consider the problem of constructing (3,L) quasi-cyclic low-density parity-check (LDPC) codes from complete protographs. A complete protograph is a small bipartite graph with two disjoint vertex sets such that every vertex in the variable-node set is connected to every vertex in the check-node set by a unique edge. This paper analyzes the required lifting factor for achieving girths of six or eight in the resulting quasi-cyclic codes with constraints on lifting. The required lifting factors provide lower bounds on the block-length of such codes.
Quasi-cyclic low-density parity-check (QC-LDPC) codes based on protographs are of great interest to code designers because analysis and implementation are facilitated by the protograph structure and the use of circulant permutation matrices for protograph lifting. However, these restrictions impose undesirable fixed upper limits on important code parameters, such as minimum distance and girth. In this paper, we consider an approach to constructing QC-LDPC codes that uses a two-step lifting procedure based on a protograph, and, by following this method instead of the usual one-step procedure, we obtain improved minimum distance and girth properties. We also present two new design rules for constructing good QC-LDPC codes using this two-step lifting procedure, and in each case we obtain a significant increase in minimum distance and achieve a certain guaranteed girth compared to one-step circulant-based liftings. The expected performance improvement is verified by simulation results.
Quasi-cyclic low-density parity-check (QC-LDPC) codes based on protographs are of great interest to code designers because analysis and implementation are facilitated by the protograph structure and the use of circulant permutation matrices for protograph lifting. However, these restrictions impose undesirable fixed upper limits on important code parameters, such as minimum distance and girth. In this paper, we consider an approach to constructing QC-LDPC codes that uses a two-step lifting procedure based on a protograph, and, by following this method instead of the usual one-step procedure, we obtain improved minimum distance and girth properties. We also present two new design rules for constructing good QC-LDPC codes using this two-step lifting procedure, and in each case, we obtain a significant increase in minimum distance and achieve a certain guaranteed girth compared with one-step circulant-based liftings. The expected performance improvement is verified by simulation results.
Quasi-Cyclic (QC) low-density parity-check (LDPC) codes are a class of LDPC codes with a simple construction facilitating hardware implementation while achieving excellent performance. In this paper, we introduce an algorithm that constructs QC spatially-coupled (SC) LDPC codes with large girth while keeping the constraint length small. The algorithm offers a "protograph to basegraph" construction, focusing on finding small lifting sizes of QC codes while avoiding short cycles. This work extends the hierarchical quasi-cyclic (HQC) construction for block LDPC codes proposed by Wang et al. to the spatially coupled case. The construction is based on the cycle relevant matrix (CRM) derived from the periodic structure of time-invariant SC-LDPC codes. Numerical results show that the proposed algorithm effectively achieves the target girth with a small lifting factor, enabling low-complexity SC code construction.