arXiv:0805.2427v1 [cs.IT] 16 May 2008 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, MAY 2008 1 On Trapping Sets and Guaranteed Error Correction Capability of LDPC Codes and GLDPC Codes Shashi Kiran Chilappagari, Student Member, IEEE, Dung Viet Nguyen, Student Mem- ber, IEEE, Bane Vasic, Senior Member, IEEE, and Michael W. Marcellin, Fellow, IEEE Abstract The relation between the girth and the guaranteed error correction capability of γ -left regular LDPC codes when decoded using the bit flipping (serial and parallel) algorithms is investigated. A lower bound on the size of variable node sets which expand by a factor of at least 3γ/4 is found based on the Moore bound. An upper bound on the guaranteed error correction capability is established by studying the sizes of smallest possible trapping sets. The results are extended to generalized LDPC codes. It is shown that generalized LDPC codes can correct a linear fraction of errors under the parallel bit flipping algorithm when the underlying Tanner graph is a good expander. It is also shown that the bound cannot be improved when γ is even by studying a class of trapping sets. A lower bound on the size of variable node sets which have the required expansion is established. Index Terms Low-density parity-check codes, bit flipping algorithms, trapping sets, error correction capability I. I NTRODUCTION Iterative algorithms for decoding low-density parity-check (LDPC) codes [1] have been the focus of research over the past decade and most of their properties are well understood [2], [3]. These algorithms operate by passing messages along the edges of a graphical representation of the code known as the Tanner graph, and are optimal when the underlying graph is a tree. Message passing decoders perform remarkably well which can be attributed to their ability to correct errors beyond the traditional bounded distance decoding capability. However, in contrast to bounded distance decoders (BDDs), the guaranteed error correction capability of iterative decoders is largely unknown. Manuscript received May 16, 2008. This work is funded by NSF under Grant CCF-0634969, ECCS-0725405, ITR-0325979 and by the INSIC-EHDR program. S. K. Chilappagari, D. V. Nguyen, B. Vasic and M. W. Marcellin are with the Department of Electrical and Computer Engineering, University of Arizona, Tucson, Arizona, 85721 USA. (emails: {shashic, nguyendv, vasic, marcellin}@ece.arizona.edu. Parts of this work have been accepted for presentation at the International Symposium on Information Theory (ISIT’08) and the International Telemetering Conference (ITC’08).