  In this paper, we propose a new characterization for elementary trapping sets
(ETSs) of variable-regular low-density parity-check (LDPC) codes. Recently,
Karimi and Banihashemi proposed a characterization of ETSs, which was based on
viewing an ETS as a layered superset (LSS) of a short cycle in the code's
Tanner graph. A notable advantage of LSS characterization is that it
corresponds to a simple LSS-based search algorithm that starts from short
cycles of the graph and finds the ETSs with LSS structure efficiently. Compared
to the LSS-based characterization of Karimi and Banihashemi, which is based on
a single LSS expansion technique, the new characterization involves two
additional expansion techniques. The introduction of the new techniques
mitigates two problems that LSS-based characterization/search suffers from: (1)
exhaustiveness, (2) search efficiency. We prove that using the three expansion
techniques, any ETS structure can be obtained starting from a simple cycle, no
matter how large the size of the structure $a$ or the number of its unsatisfied
check nodes $b$ are. We also demonstrate that for the proposed
characterization/search to exhaustively cover all the ETS structures within the
$(a,b)$ classes with $a \leq a_{max}, b \leq b_{max}$, for any value of
$a_{max}$ and $b_{max}$, the length of the short cycles required to be
enumerated is less than that of the LSS-based characterization/search. We also
prove that the three expansion techniques, proposed here, are the only
expansions needed for characterization of ETS structures starting from simple
cycles in the graph, if one requires each and every intermediate sub-structure
to be an ETS as well. Extensive simulation results are provided to show that,
compared to LSS-based search, significant improvement in search speed and
memory requirements can be achieved.
