KKTCode: Asymptotically Optimal Linear Codes for Noisy Feedback AWGN Channels

2026-09-26

AI Summary

This study addresses the long-standing, six-decade open problem of optimal causal linear coding over additive white Gaussian noise (AWGN) channels with noisy feedback. By leveraging Karush–Kuhn–Tucker (KKT) optimization theory and polynomial eigenvalue analysis, this work proposes an active linear feedback coding framework that transcends the theoretical limitations of conventional passive schemes. Furthermore, it reveals underlying geometric Toeplitz structural properties to design a highly efficient algorithm with logarithmic complexity. The proposed approach rigorously establishes global optimality in terms of mean squared error (MSE) and signal-to-noise ratio (SNR), thereby unifying active and passive solution paradigms. Theoretical analysis demonstrates that the scheme asymptotically achieves the Elias–Butman information-theoretic performance limit, while numerical experiments validate both its optimality and computational efficiency.

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.

Problem

• AWGN channels
• noisy feedback
• linear codes
• causal feedback
• active feedback

Innovation

• Noisy Feedback AWGN Channels
• Active Linear Feedback
• KKT-optimal Design
• Geometric Toeplitz Structure
• Causal Linear Codes