Variable-Length Coding for Binary-Input Channels With Limited Stop Feedback
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This paper focuses on the numerical evaluation of the maximal achievable rate of variable-length stop-feedback (VLSF) codes with m decoding times at a given message size and error probability for binary-input additive white Gaussian noise channel, binary symmetric channel, and binary erasure channel (BEC). Leveraging the Edgeworth and Petrov expansions, we develop tight approximations to the tail probability of length-n cumulative information density that are accurate for any blocklength n. We reduce Yavas et al.'s non-asymptotic achievability bound on VLSF codes with m decoding times to an integer program of minimizing the upper bound on the average blocklength subject to the average error probability, minimum gap, and integer constraints. We develop two distinct methods to solve this program. Numerical evaluations show that Polyanskiy's achievability bound for VLSF codes, which assumes m = ∞, can be approached with a relatively small m in all of the three channels. For BEC, we consider systematic transmission followed by random linear fountain coding. This allows us to obtain a new achievability bound stronger than a previously known bound and new VLSF codes whose rate further outperforms Polyanskiy's bound.
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