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}.

翻译：本文针对AR($p$)高斯信道提出了一种广义化著名Schalkwijk-Kailath (SK) 编码方案的高斯随机编码方案。该构造性编码方案称为SK(2)编码方案，可为相应可达速率提供闭式表征。该结果特别表明，著名的SK编码方案并非普遍最优，从而推翻了Butman在\cite{butman1976linear}中提出的猜想。