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Some Derived Results for Constraints on the Tanner Graph Girth of Quasi-Cyclic Ldpc Codes

Huaan Li, Wenzhen Wang, Yunfei Fang, Hengzhou Xu

Published Jan 1, 2024• 0 citations
Publisher
DOI
10.2139/ssrn.4941917

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In this paper, we study the relationships between the girth of the Tanner graph of a quasi cyclic (QC) protograph low-density parity-check (LDPC) code, on one hand, and the lifting degree and the size and the structure of the base graph, on the other hand. As a result, for a given base graph and a given lifting degree, we derive an upper bound on the girth of the resulting lifted graphs (codes). The upper bounds derived here are generally tighter than the existing bounds. The results presented in this work can be used to select an appropriate lifting degree for a given base graph, in order to have a desired girth, or to provide some insight in designing good base graphs, or to properly select the base graph's edge permutations.
In this paper, we study the relationships between the girth of the Tanner graph of a quasi-cyclic (QC) protograph low-density parity-check (LDPC) code, the lifting degree, and the size and the structure of the base graph. As a result, for a given base graph, we derive a lower bound on the lifting degree as a necessary condition for the lifted graph to have a certain girth. This also provides an upper bound on the girth of the family of graphs lifted from a given base graph with a given lifting degree. The upper bounds derived here, which are applicable to both regular and irregular base graphs with no parallel edges, are in some cases more general and in some other cases tighter than the existing bounds. The results presented in this work can be used to design cyclic liftings with relatively small degree and relatively large girth. As an example, we present new QC protograph LDPC code constructions with girth 8 using fully connected base graphs. These constructions provide upper bounds on the lifting degree required for achieving girth 8 using fully connected base graphs.
In this letter, we propose a new technique to design quasi-cyclic low-density parity-check (QC-LDPC) codes with girth 8 and variable node degrees 3 and 4. Small trapping sets are avoided in the designed codes by imposing simple constraints on the cycles of length 8 in the code's Tanner graph. We also derive lower bounds on the block length of the codes that can satisfy such constraints. Simulation results are provided to demonstrate the superiority of the designed codes in comparison with similar codes in the literature.