Entropy of Bernoulli Measures Conditioned on Affine Subspaces and a Problem of Ancheta-Massey

For a Bernoulli$(p)$ source with $0<p<\frac12$, the paper proves that the best lossy compression achievable by a linear encoder is exactly the simple time-sharing strategy that losslessly compresses a fraction of the bits and estimates the rest as zero. Equivalently, every full-row-rank $H\in\mathbb F_2^{k\times n}$ satisfies $$\frac{k}{n}\ge h(p)\left(1-\frac{D(H)}p\right).$$ This resolves Massey's question affirmatively for all $p<\frac12$; Ancheta had already proved the $p=\frac12$ case.

For a Bernoulli$(p)$ source with $0<p<\frac12$, the paper proves that the best lossy compression achievable by a linear encoder is exactly the simple time-sharing strategy that losslessly compresses a fraction of the bits and estimates the rest as zero. Equivalently, every full-row-rank $H\in\mathbb F_2^{k\times n}$ satisfies $$\frac{k}{n}\ge h(p)\left(1-\frac{D(H)}p\right).$$ This resolves Massey's question affirmatively for all $p<\frac12$; Ancheta had already proved the $p=\frac12$ case.