Fixed Initialization Decoding of LDPC Codes Over a Binary Symmetric Channel

We introduce in this paper the concept of a correctable error set and a fixed initialization decoding, by noticing that the sum-product decoder with a given iteration number only depends on the initialized probability of error, for a BSC. Although this value has been conventionally selected as the BSC crossover probability, we show that other selections can provide better performance or faster convergence. We also prove that for any fixed initialization (i.e., any given correctable error set), the word-error-rate can be represented as a polynomial of the BSC crossover probability. This suggests that the word-error-rate can be analytically derived from the knowledge of the correctable error set.