LDPC LCD codes from odd graphs

In this paper, we study binary low-density parity-check, or LDPC codes, determined by the adjacency matrices of the odd graphs as their parity-check matrices. We show that for the odd graph $ O_n $, $ n\geq 3 $, the obtained code $ C_n $ is an $ (n, n) $-regular binary LDPC code of length $ {2n-1 \choose n-1} $, dimension $ {2n-2 \choose n-2} $ and minimum distance $ n+1 $. Further, we show that the girth of the corresponding Tanner graph determined by the adjacency matrix of an odd graph is equal to 6, and give the expression for the variance of the syndrome weight for $ C_n $. Moreover, we show that there are no absorbing sets of size smaller than 3 for these codes and that the absorbing sets of size 3 exist only for $ n = 3 $. Additionally, we give the structure of $ (a, b) $ absorbing sets with $ b \leq a\leq n $. Finally we prove that $ C_n $ is also a linear code with complementary dual, or LCD code, for $ n\geq3 $.