AI 中文总结
该研究将Ancheta的伯努利测度熵结论推广至所有$p<\ rac{1}{2}$的情况，证明由GPT-5.6 Sol发现，旨在建立其与编码理论及自旋玻璃理论的联系。
AI 中文摘要
信息论中的一个经典结论是，线性编码器可实现参数为$p$的伯努利信源无损压缩的熵。然而在有损压缩场景中，线性编码相比率失真函数存在严格的次优性。Massey提出疑问：线性编码的最优速率是否可通过线性无损压缩一部分比特，其余比特置零估计来实现。当$p=\ rac{1}{2}$时，Ancheta给出了肯定回答。本注将Ancheta的结果推广至所有$p<\ rac{1}{2}$的情况，核心论证是基于仿射子空间的后验分布熵以其边缘分布为界。该证明由GPT-5.6 Sol在作者指导的交互过程中发现，本注的目的是传播该证明的简化版本，并建立其与现有编码理论及自旋玻璃理论文献的联系。
英文摘要
A textbook result in information theory is that linear encoders achieve the entropy for lossless compression of Bernoulli source with parameter $p$. For lossy compression, however, linearity is known to incur strict suboptimality compared to the rate-distortion function. Massey asked whether the optimal rate for linear encoding is achieved simply by compressing a fraction of the bits linearly and losslessly and estimating the rest by zero \cite{Massey1978}. For $p=\frac12$, Ancheta answered this question affirmatively \cite{Ancheta1978}. This note extends Ancheta's result to all $p<\frac12$. The key argument is to bound the entropy of the posterior distribution conditioned on an affine subspace in terms of its marginals. The proof was discovered by GPT-5.6 Sol in an interactive process guided by the author. The purpose of the present note is to communicate a simplified version of this proof and to make connections with the existing literature on coding theory and spin glass theory.
