LDPC Codes Which Can Correct Three Errors Under Iterative Decoding
  In this paper, we provide necessary and sufficient conditions for a column-weight-three LDPC code to correct three errors when decoded using Gallager A algorithm. We then provide a construction technique which results in a code satisfying the above conditions. We also provide numerical assessment of code performance via simulation results. 
Shashi Kiran Chilappagari, Anantha Raman Krishnan, Bane Vasic