KKTCode: Asymptotically Optimal Linear Codes for Noisy Feedback AWGN Channels
Abstract
The design of optimal causal linear feedback schemes for additive white Gaussian noise (AWGN) channels with noisy output feedback has remained an open problem for over 60 years. Prior work has focused on restricted policy classes, especially passive (uncoded) noisy output feedback, where only the transmitter performs feedback coding. However, passive noisy output feedback fundamentally lacks the degrees of freedom required to attain the information-theoretic performance limit in general. In this paper, we consider the active (coded) noisy output feedback setting, where both the transmitter and the receiver perform feedback coding. We then develop a constructive KKT-optimal active linear feedback design that asymptotically attains the Elias-Butman SNR converse bound, thereby establishing MSE/SNR optimality over the entire class of causal linear schemes. Furthermore, we prove that the optimal passive feedback solution is recovered as a special case of the active design. This passive solution admits a Geometric Toeplitz (GT) structure with a Chance-Love (CL)-style one-shot polynomial characterization, and can be computed with O(log T) complexity. Thus, our results provide an affirmative answer to the long-standing optimality question for noisy output feedback under causal linear feedback coding, and our numerical results support the theoretical findings.
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BibTeX
@misc{nam2026kktcode,
title = {{KKTCode: Asymptotically Optimal Linear Codes for Noisy Feedback AWGN Channels}},
author = {Hongjae Nam and Jonggyu Jang and Vishrant Tripathi and David J. Love},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.32118},
eprint = {2609.32118},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.32118},
}
RIS
TY - UNPB
TI - KKTCode: Asymptotically Optimal Linear Codes for Noisy Feedback AWGN Channels
AU - Hongjae Nam
AU - Jonggyu Jang
AU - Vishrant Tripathi
AU - David J. Love
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2609.32118
UR - https://arxiv.org/abs/2609.32118
AB - The design of optimal causal linear feedback schemes for additive white Gaussian noise (AWGN) channels with noisy output feedback has remained an open problem for over 60 years. Prior work has focused on restricted policy classes, especially passive (uncoded) noisy output feedback, where only the transmitter performs feedback coding. However, passive noisy output feedback fundamentally lacks the degrees of freedom required to attain the information-theoretic performance limit in general. In this paper, we consider the active (coded) noisy output feedback setting, where both the transmitter and the receiver perform feedback coding. We then develop a constructive KKT-optimal active linear feedback design that asymptotically attains the Elias-Butman SNR converse bound, thereby establishing MSE/SNR optimality over the entire class of causal linear schemes. Furthermore, we prove that the optimal passive feedback solution is recovered as a special case of the active design. This passive solution admits a Geometric Toeplitz (GT) structure with a Chance-Love (CL)-style one-shot polynomial characterization, and can be computed with O(log T) complexity. Thus, our results provide an affirmative answer to the long-standing optimality question for noisy output feedback under causal linear feedback coding, and our numerical results support the theoretical findings.
ER -