Low-Density Parity-Check Codes Which Can Correct Three Errors Under   Iterative Decoding
  In this paper, we give necessary and sufficient conditions for low-density parity-check (LDPC) codes with column-weight three to correct three errors when decoded using hard-decision message-passing decoding. Additionally, we give necessary and sufficient conditions for column-weight-four codes to correct three errors in four iterations of hard-decision message-passing decoding. We then give a construction technique which results in codes satisfying these conditions. We also provide numerical assessment of code performance via simulation results. 
Shashi Kiran Chilappagari, Anantha Raman Krishnan, Bane Vasic, Michael   W. Marcellin