Extremal Problems of Information Combining

Abstract

Abstract-In this paper, we study moments of soft bits of binary-input symmetric-output channels and solve some extremal problems of the moments. We use these results to solve the extremal information combining problem. Further, we extend the information combining problem by adding a constraint on the second moment of soft bits, and find the extremal distributions for this new problem. The results for this extension problem are used to improve the prediction of convergence of the belief propagation decoding of low-density parity-check (LDPC) codes, provided that another extremal problem related to the variable nodes is solved.Index Terms-Extrinsic information transfer (EXIT) functions, information combining, low-density parity-check (LDPC) codes.

On Universal Properties of Capacity-Approaching LDPC Code Ensembles

“…Remark 10: This lemma is in fact equivalent to the statement in [20,Theorem 1] with the extreme values derived in its proof (note that (20) implies that the sequence {g k } is equal to the sequence {m 2k } in [20], from which the equivalence between Lemma 5 and [20, Theorem 1] follows directly). In Appendix II, we present an alternative proof which is more elementary.…”

“…Note that for a BEC with erasure probability p, g k = 1 − p for all k ∈ N (in this case we have L ∈ {0, +∞} with probabilities p and 1 − p, respectively, and the equality tanh(+∞) = 1 is exploited in (20)). Therefore (39) is particularized to…”

“…The proof of the lower bound on g 1 relies on (94). Since the sequence {g k } k≥1 is monotonically non-increasing and non-negative (this property follows directly from (20)), then…”

“…Eq. (20) implies that the non-negative sequence {g p } p≥1 is monotonically non-increasing and it only depends on the communication channel (but not on the code). Note also that, from (20), 0 ≤ g p < 1 for all p ∈ N (unless the channel is perfect, which then implies that g p = 1 for all values of p).…”

“…(20) implies that the non-negative sequence {g p } p≥1 is monotonically non-increasing and it only depends on the communication channel (but not on the code). Note also that, from (20), 0 ≤ g p < 1 for all p ∈ N (unless the channel is perfect, which then implies that g p = 1 for all values of p). We note that the conditional entropy on the LHS of (18) depends only on the code and the communication channel, but its lower bound on the RHS of (18) depends also on the specific representation of the code by a bipartite graph.…”