Interactive Encoding and Decoding Based on Binary LDPC Codes with Syndrome Accumulation

Interactive encoding and decoding based on binary low-density parity-check codes with syndrome accumulation (SA-LDPC-IED) is proposed and investigated. Assume that the source alphabet is $\mathbf{GF}(2)$, and the side information alphabet is finite. It is first demonstrated how to convert any classical universal lossless code $\mathcal{C}_n$ (with block length $n$ and side information available to both the encoder and decoder) into a universal SA-LDPC-IED scheme. It is then shown that with the word error probability approaching 0 sub-exponentially with $n$, the compression rate (including both the forward and backward rates) of the resulting SA-LDPC-IED scheme is upper bounded by a functional of that of $\mathcal{C}_n$, which in turn approaches the compression rate of $\mathcal{C}_n$ for each and every individual sequence pair $(x^n,y^n)$ and the conditional entropy rate $\mathrm{H}(X |Y)$ for any stationary, ergodic source and side information $(X, Y)$ as the average variable node degree $\bar{l}$ of the underlying LDPC code increases without bound. When applied to the class of binary source and side information $(X, Y)$ correlated through a binary symmetrical channel with cross-over probability unknown to both the encoder and decoder, the resulting SA-LDPC-IED scheme can be further simplified, yielding even improved rate performance versus the bit error probability when $\bar{l}$ is not large. Simulation results (coupled with linear time belief propagation decoding) on binary source-side information pairs confirm the theoretic analysis, and further show that the SA-LDPC-IED scheme consistently outperforms the Slepian-Wolf coding scheme based on the same underlying LDPC code. As a by-product, probability bounds involving LDPC established in the course are also interesting on their own and expected to have implications on the performance of LDPC for channel coding as well.