On the capacity of the binary symmetric relay channel with a finite memory relay

We investigate capacity bounds for two instances of the binary symmetric relay channels. For one of these, we establish the capacity, and for the other one we present various capacity bounds. We first evaluate the cut-set bound and the rates achievable using decode-and-forward (DF) and compress-and-forward (CF) relaying. We then focus on relaying strategies with finite memory-length and present an efficient algorithm for optimizing the relay functions. We use the boolean Fourier transform to unveil the structure of the optimized mappings. Interestingly, the optimized relay functions exhibit a simple structure. The numerical results show that the rates achieved using the optimized low-dimensional functions are either comparable or superior to those achieved by DF and CF relaying. In particular, the optimized low-dimensional relaying scheme can improve on DF relaying when the quality of the source-relay link is worse than or comparable to the quality of other links.