A Recursive Method to Construct the Optimal Exponent Matrices

A Recursive Method to Construct the Optimal Exponent Matrices

Recently, an exhaustive search has been used to find all of the possible non-isomorphic 4-cycle free column-weight three QC-LDPC codes with the shortest length. The drawback of this approach, however, is the complexity which increases sharply by extending the exponent matrices. Moreover, the minimum lifting degrees of these codes are not the same with the lower-bounds, in general, i.e. QC-LDPC codes with the same exponent matrix may have cycles of length 4, for some lifting degrees greater than the given shortest length. Here, an explicit ordering on the positions of the desired exponent matrix is proposed to construct some column-weight three QC-LDPC codes with girth 6 having the shortest length. The constructed codes have two main benefits: The lower-bound is the same with the minimum-lifting degree and the overall complexity is polynomial, in terms of the length of the constructed codes.

Recently, an exhaustive search has been used to find all of the possible non-isomorphic 4-cycle free column-weight three QC-LDPC codes with the shortest length. The drawback of this approach, however, is the complexity which increases sharply by extending the exponent matrices. Moreover, the minimum lifting degrees of these codes are not the same with the lower-bounds, in general, i.e. QC-LDPC codes with the same exponent matrix may have cycles of length 4, for some lifting degrees greater than the given shortest length. Here, an explicit ordering on the positions of the desired exponent matrix is proposed to construct some column-weight three QC-LDPC codes with girth 6 having the shortest length. The constructed codes have two main benefits: The lower-bound is the same with the minimum-lifting degree and the overall complexity is polynomial, in terms of the length of the constructed codes.