On decoding failure probabilities for linear block codes on the binary erasure channel

It has been claimed that the performance of a linear block code under iterative decoding on the binary erasure channel is determined by the stopping distance, Le, the size of the smallest non-empty stopping set in the associated Tanner graph. Indeed, this is true from the perspective of code word retrieval. However, we show that with respect to the retrieval of just the information bits within the code word, the stopping distance may not be the main performance indicator since the smallest non-empty stopping sets might not hit the information set. We present expressions for decoding failure probabilities for code/information words/bits under optimal or iterative decoding and demonstrate the importance of the choice of the set of bits carrying the information for a fixed code with a fixed decoder.