This paper is concerned with iterative decoding of short error correcting codes having high density parity check matrices (HDPC Codes). There is a need for reliable graph-based iterative decoding of short codes; for instance this is beneficial in the framework of a newly introduced random version of OFDMA (R-OFDMA) . We first note how the structure of the parity check matrix can alter an iterative decoder's performance. Specifically we focus on the allowable redundancies in the matrices, where it is possible to benefit from inserting redundant parity checks without altering the code properties. We then employ computation tree analysis and density evolution which both support simulation results showing a performance improvement.