Invariance Properties of Binary Linear Codes Over a Memoryless Channel With Discrete Input

This work studies certain properties of the probability density function (pdf) of the bit log-likelihood ratio (LLR) for binary linear block codes over a memoryless channel with discrete input and discrete or continuous output. We prove that under a set of mild conditions, the pdf of the bit LLR of a specific bit position is independent of the transmitted codeword. 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 vertical axis). For the case of channels with binary input, a sufficient condition for two bit positions to have the same pdf is presented