On the Smallest Absorbing Sets of LDPC Codes From Finite Planes

Absorbing sets, a class of combinatorial structures of the Tanner graph representation of a low-density parity-check (LDPC) code, are known to influence the performance of the code under message passing iterative decoding. In this paper, we study the smallest absorbing sets of LDPC codes constructed from projective planes and Euclidean planes. The lower bounds on the parameters of smallest absorbing sets given by Dolecek are proven to be tight for these two families of LDPC codes. We also analyze the combinatorial properties of the smallest absorbing sets and give conditions necessary and sufficient for a set of bit nodes in the Tanner graph to be a smallest absorbing set. For LDPC codes from projective planes, we further give a condition necessary and sufficient for a smallest absorbing set to be a fully absorbing set. In addition, we show that these smallest absorbing sets are asymptotically not stable, which may explain to some extent the good performance as well as the low error floor expectation of these two families of LDPC codes.