Probability of even parity of soft bits

Abstract

Gallager's lemma, presented originally in the context of decoding low-density parity check (LDPC) codes, computes the probability of even parity for independent, non-identically distributed bits. As the number of bits involved in the parity increases, there is a tendency of the probability of even parity towards 1/2. This tendency is discussed and analysed, resulting in a sort of counter central limit theorem result. In many instances, there are practical limits on the number of bits that can be included in a reliable soft parity computation. Implications affect LDPC decoding, soft descrambling and soft encoding of data.