On ML redundancy of codes

The ML redundancy of a code is defined as the smallest number of rows in its parity-check matrix such that a message-passing decoder working in the corresponding Tanner graph achieves maximum-likelihood (ML) performance on an erasure channel. General upper bounds on ML redundancy are obtained. In particular, it is shown that the ML redundancy of a q-ary code is at most the number of minimal codewords in its dual code, divided by q-1. Special upper bounds are derived for codes whose dual code contains a covering design. For example, the ML redundancy of a Simplex code of length n is shown to be no greater than (n² - 4n + 9)/6.