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In this paper, we give necessary and sufficient conditions for low-densityparity-check (LDPC) codes with column-weight three to correct three errors whendecoded using hard-decision message-passing decoding. Additionally, we givenecessary and sufficient conditions for column-weight-four codes to correctthree errors in four iterations of hard-decision message-passing decoding. Wethen give a construction technique which results in codes satisfying theseconditions. We also provide numerical assessment of code performance viasimulation results.

Low-Density Parity-Check Codes Which Can Correct Three Errors Under Iterative Decoding