Disjoint Difference Sets and QC-LDPC Codes With Girth 10

A method to obtain an exponent matrix of a QCLDPC code with column weight 4 appeared recently in which the entries of the matrix belong to the sets of a (v,k,2) disjoint difference set (DDS). It was also shown that the girth of the Tanner graph in these codes is at least 8. In this paper, we prove the existence of an 8-cycle in their Tanner graph which shows that the girth of these codes is exactly 8.The merits of a QC-LDPC code from a DDS motivated us to study these combinatorial designs. However, the disjoint difference sets existing in the literature are limited to small values of the parameters k and t. We present a novel approach to obtain a (v,k,t)-DDS with 3 ≤ t ≤ 5 and different values of k. Our method is based on a new relation between QC-LDPC codes with girth 10 and disjoint difference sets.