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高斯随机编码方案，该方案推广了著名的Schalkwijk-Kailath（SK）编码方案。这种构造性编码方案称为SK(2)编码方案，能够为相应的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在文献\cite{butman1976linear}中提出的猜想。