Probabilistic Decoding of Majority Codes

The transmission of binary messages over a memoryless channel with a continuous output is considered. For binary codes which allow majority decoding a decision scheme is used which is optimal with respect to the maximum a posteriori probability and uses the a posteriori error probabilities of the received symbols. These probabilities give different weights to the checks, which allows optimal reception as a whole to be approached. A modification of the optimal algorithm which has a simpler realization is likewise described. It is shown that for weak Gaussian noise the use of probabilistic decoding is equivalent to increasing the number of checks for symbol-wise reception by approximately a factor of two.