  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.
