Check-hybrid GLDPC codes without small trapping sets

In this paper, we propose a new approach to construct a class of check-hybrid generalized low-density parity-check (GLDPC) codes which are free of small trapping sets. The approach is based on converting some selected check nodes involving a trapping set to super checks corresponding to a shorter error-correcting component code. Specifically, we follow two main purposes to construct the check-hybrid codes: First, replacing single parity checks by super checks is done based on the knowledge of the trapping sets of the global LDPC code. We show that by converting some single checks to super checks in a trapping set, the decoder corrects the errors on a trapping set and hence eliminates the trapping set. Second, the rate-loss caused by replacing the super checks is reduced through finding the minimum number of such critical checks. We first present an algorithm to find possible critical checks in a trapping set. We then provide some upper bounds on the minimum number of such critical checks such that the decoder corrects all error patterns on certain trapping sets in the Tanner graph of the global LDPC code. We also provide a potential fixed set for a class of constructed check-hybrid codes.