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This paper presents new achievability bounds on the maximal achievable rate of variable-length stop-feedback (VLSF) codes operating over a binary erasure channel (BEC) at a fixed message size M=2^k . We provide bounds for two cases: The first case considers VLSF codes with possibly infinite decoding times and zero error probability. The second case limits the maximum (finite) number of decoding times and specifies a maximum tolerable probability of error. Both new achievability bounds are proved by constructing a new VLSF code that employs systematic transmission of the first k message bits followed by random linear fountain parity bits decoded with a rank decoder. For VLSF codes with infinite decoding times, our new bound outperforms the state-of-the-art result for BEC by Devassy et al. in 2016. We show that the backoff from capacity reduces to zero as the erasure probability decreases, thus giving a negative answer to the open question Devassy et al. posed on whether the 23.4% backoff to capacity at k=3 is fundamental to all BECs. For VLSF codes with finite decoding times, numerical evaluations show that the systematic transmission followed by random linear fountain coding performs better than random linear coding in terms of achievable rates.
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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.
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- Award ID(s):
- 1955660
- PAR ID:
- 10398505
- Date Published:
- Journal Name:
- 2022 IEEE International Symposium on Information Theory (ISIT)
- Page Range / eLocation ID:
- 2327 to 2332
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1109/ISIT50566.2022.9834526
