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One of the most significant impediments to the use of LDPC codes in many communication and storage systems is the error-rate floor phenomenon associated with their iterative decoders. The error floor has been attributed to certain sub-graphs of an LDPC code's Tanner graph induced by so-called trapping sets. We show in this paper that once we identify the trapping sets of an LDPC code of interest, a sum-product algorithm (SPA) decoder can be custom-designed to yield floors that are orders of magnitude lower than the conventional SPA decoder. We present two classes of such decoders: a bi-mode syndrome-erasure decoder and three generalized-LDPC decoders. We demonstrate the effectiveness of these decoders for two codes, the rate-1/2 (2640,1320) Margulis code which is notorious for its floors and a rate-0.3 (640,192) quasi-cyclic code which has been devised for this study.

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