Variable-Length Stop-Feedback Codes With Finite Optimal Decoding Times for BI-AWGN Channels

In this paper, we are interested in the performance of a variable-length stop-feedback (VLSF) code with m optimal decoding times for the binary-input additive white Gaussian noise channel. We first develop tight approximations to the tail probability of length-n cumulative information density. Building on the work of Yavas et al., for a given information density threshold, we formulate the integer program of minimizing the upper bound on average blocklength over all decoding times subject to the average error probability, minimum gap and integer constraints. Eventually, minimization of locally optimal upper bounds over all thresholds yields the globally minimum upper bound and the above method is called the two-step minimization. Relaxing to allow positive real-valued decoding times activates the gap constraint. We develop gap-constrained sequential differential optimization (SDO) procedure to find the optimal, gap-constrained, real-valued decoding times. In the error regime of practical interest, Polyanskiy's scheme of stopping at zero does not help. In this region, the achievability bounds estimated by the two-step minimization and gap-constrained SDO show that Polyanskiy’s achievability bound for VLSF codes can be approached with a small number of decoding times.