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Journal of Applied Sciences

Construction of High Girth and Two Column Weight LDPC Code Based on Graph

Mahnaz Ahmadi, Hadi Dehghani, ‎Saeid Alikhani, Roslan Hasni

Published Apr 1, 2012• 1 citation
Publisher
DOI
10.3923/jas.2012.798.801

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LDPC codes of column weight of two are constructed from minimal distance graphs or cages. Distance graphs are used to represent LDPC code matrices such that graph vertices that represent rows and edges are columns. The conversion of a distance graph into matrix form produces an adjacency matrix with column weight of two and girth double that of the graph. The number of 1's in each row (row weight) is equal to the degree of the corresponding vertex. By constructing graphs with different vertex degrees, we can vary the rate of corresponding LDPC code matrices. Cage graphs are used as examples of distance graphs to design codes with different girths and rates. Performance of obtained codes depends on girth and structure of the corresponding distance graphs.
An approach to construct column-weight-2 LDPC codes with high girth is presented. The approach derives a column-weight-2 LDPC code from the Tanner graph of a (k,k) QC-LDPC code. By the construction, the new LDPC codes double in girth. To construct the (k,k) QC-LDPC codes with desired girth, a search algorithm is proposed in this paper. The approach generates an example of a (2,3) LDPC code with a girth of 36, which is larger than the column-weight-2 LDPC codes constructed by the previously methods.
In this letter, an approach is proposed to increase the row (column)-weight of the parity-check matrix of a 4-cycle free LDPC code such that the constructed LDPC code has girth of at least 6. In fact, to each parity-check matrix H, a new graph Gr(H) (Gc(H)) is assigned in which the vertices correspond to the rows (columns) of H and two vertices are adjacent if and only if the associated rows (columns) have in common at least one nonzero element. Now, in a proper vertex coloring of Gr(H) (Gc(H)), each color is considered as a new column (row) whose nonzero elements happen in the rows (columns) in which the corresponding vertices have the same color. Based on this method, some high-rate LDPC codes with girth 6 and column-weight of at least 4 can be constructed from the recently proposed column-weight three LDPC codes with girth 6. Moreover, using the mutually orthogonal Latin squares, this approach is applied on the incidence matrices of some complete bipartite graphs to generate some girth-6 LDPC codes with different column-weights and large minimum-distances.