The Approximate Maximum-Likelihood Certificate

The confidence in the reliability of a codeword output by some (not necessarily optimal) decoding algorithm is discussed. A new property which relies on the linear programming (LP) decoder, the approximate maximum-likelihood certificate (AMLC), is introduced to address this issue as follows. First, the channel output vector is decoded by some symmetric decoder D, e.g., belief propagation or min-sum algorithm decoding. Second, the channel output vector is decoded by LP decoding. Third, if the decoding result of D is a codeword, its LP value is compared to the LP value of the LP decoding result (the latter need not be a codeword). If these two values are close, the AMLC holds. Using upper bounding techniques, we show that the conditional frame error probability given that the AMLC holds, is with some degree of confidence below a threshold. In channels with low noise, this threshold is orders of magnitude lower than the simulated frame error rate, and our bound holds with a very high degree of confidence. This is in stark contrast with standard Monte Carlo simulation, which would require excessively long runs to demonstrate like performance. When the AMLC holds, our approach thus provides the decoder with extra error detection capability, which is especially important in applications requiring high data integrity.