Hamming codes as error-reducing codes

Hamming codes are the first nontrivial family of error-correcting codes that can correct one error in a block of binary symbols. In this paper we extend the notion of error-correction to error-reduction and present several decoding methods with the goal of improving the error-reducing capabilities of Hamming codes. First, the error-reducing properties of Hamming codes with standard decoding are demonstrated and explored. We show a lower bound on the average number of errors present in a decoded message when two errors are introduced by the channel for general Hamming codes. Other decoding algorithms are investigated experimentally, and it is found that these algorithms improve the error reduction capabilities of Hamming codes beyond the aforementioned lower bound of standard decoding.