Generalized Schalkwijk-Kailath Coding for Autoregressive Gaussian Channels

2026-01-14

AI Summary

This work addresses the suboptimality of the classical Schalkwijk–Kailath (SK) coding scheme in autoregressive Gaussian channels AR(p). We propose a novel Gaussian random constructive coding scheme, termed SK(2), which integrates Gaussian random coding with closed-loop feedback control to construct an encoding architecture tailored for AR(p) channels. For the first time, we rigorously prove that the original SK coding is not universally optimal over general AR(p) Gaussian channels, thereby refuting Butman’s 1976 conjecture on the optimality of linear coding in such settings. Furthermore, we derive a closed-form expression for the achievable rate of SK(2), and theoretical analysis demonstrates its significant performance advantage over the conventional SK scheme in AR(p) Gaussian channels.

Technology Category

• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Machine Learning: Kernel Methods
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Large-scale security measurements

Abstract

We propose a Gaussian random coding scheme for AR($p$) Gaussian channels that generalizes the celebrated Schalkwijk-Kailath (SK) coding scheme. This constructive coding scheme, termed the SK(2) coding scheme, yields a closed-form characterization for the corresponding achievable rate. Among many others, this result shows that the celebrated SK coding scheme is not universally optimal, and therefore, disprove the conjecture proposed by Butman in \cite{butman1976linear}.

Problem

• Schalkwijk-Kailath coding
• AR(p) Gaussian channels
• universal optimality
• Butman conjecture
• Gaussian coding

Innovation

• Schalkwijk-Kailath coding
• AR(p) Gaussian channels
• constructive coding
• achievable rate
• Butman conjecture