Ancheta–Massey problem

Let $X^n=(X_1,5e5c,5e5c,X_n)$ have independent coordinates distributed as $\operatorname{Bernoulli}(p)$, with $0<p<\tfrac12$, and let distortion be normalized Hamming distortion. Define $R_{\mathrm{lin},p}(D)$ as the infimum of the asymptotic rates $\limsup_{n\to\infty}m_n/n$ over schemes whose encoder $f_n:\mathbb{F}_2^n\to\mathbb{F}_2^{m_n}$ is linear, whose decoder is arbitrary, and whose expected normalized Hamming distortion is at most $D$. The Ancheta–Massey conjecture asserts that $

Equivalently, the optimal linear scheme is obtained by encoding a fraction $1-D/p$ of the source bits losslessly and reproducing the remaining fraction as zero. The claim for $p=\tfrac12$ is due to Ancheta; the extension to every $p<\tfrac12$ is the problem addressed by the supplied note.