On Characterization and Efficient Exhaustive Search of Elementary Trapping Sets of Variable-Regular LDPC Codes

Recently, Karimi and Banihashemi demonstrated that a large majority of the elementary trapping set (ETS) structures of variable-regular low-density parity-check (LDPC) codes are layered supersets (LSS) of short cycles. The LSS property corresponds to a simple search algorithm that can find all ETSs with LSS structure starting from short cycles in a guaranteed fashion. In this letter, we complement this characterization by demonstrating that the remaining structures of ETSs, that are not LSS of short cycles, are all LSS of a small number of other graphical structures within the Tanner graph of the code, and thus can also be found efficiently. This together with the results of Karimi and Banihashemi provides a simple search algorithm that can find all the (a,b) ETSs of any variable-regular LDPC code for any size a and any number of unsatisfied check nodes b in a guaranteed fashion.