Girth of the Tanner graph and error correction capability of LDPC codes

We investigate the relation between the girth and the guaranteed error correction capability of γ-left regular LDPC codes. For column-weight-three codes, we give upper and lower bounds on the number of errors correctable by the Gallager A algorithm. For higher column weight codes, we find the number of variable nodes which are guaranteed to expand by a factor of at least 3γ/4, hence giving a lower bound on the guaranteed correction capability under the bit flipping (serial and parallel) algorithms. We also establish upper bounds by studying the sizes of smallest possible trapping sets.