Some Properties of Bit Decoding Algorithms for Binary Linear Block Codes

In this paper, we study certain properties of the bit decoding algorithms for the case of binary linear block codes. Our focus is on the Probability Density Function (pdf) of the bit Log-Likelihood-Ratio (LLR). A general channel model with discrete input and discrete or continuous output is considered. We prove that under a set of mild conditions on the channel, the pdf of the bit LLR of a specific bit position is independent of the transmitted code-word. It is also shown that the pdf of a given bit LLR when the corresponding bit takes the values of zero and one are symmetric with respect to each other (reflection of one another with respect to the origin). For the case of channels with binary input, a sufficient condition for two bit positions to have the same pdf is presented.