Following the proof of Lemma 25, let \(T_\ell \sim \mathcal{T}_\ell \), \(T_r\sim \mathcal{T}_r\) be the numbers of bits trimmed off the left and right of the output \(Y=\mathrm{DC}\, X\) (with \(X\) on \(\{ 0,1\} ^m\)), and set \(L=T_\ell \), \(R=m-T_r\). We prove the analogue of Claim 26 with the modified second part

\(Y'=\mathrm{TDC}_{\mathcal{T}_\ell ,\mathcal{T}_r}\, X\). The proof of the first equation is identical to Claim 26. For the second, by the same chain-rule argument (Lemma 12) we reach \(I(X;Y\mid \widetilde L,\widetilde R)\le I(X;Y_{L+1}^{R-1}\mid \widetilde L, \widetilde R)+H(Y_1^L,Y_R^m)\), but now the trimmed ends are no longer all-zero, so we can only bound \(H(Y_1^L,Y_R^m)\le \log ((2^{m\nu \eta })^2)=2m\nu \eta \). The rest of the argument proceeds as in Lemma 25, giving the stated bound.

Let \(Y^1,\dots ,Y^t\) be the outputs of \(\mathrm{DC}\) on the individual bits of \(c\) fully contained in the first \(\nu \eta m\) output bits. Up to \(o(m)\) leftover bits (negligible with high probability), the window equals \(Y^1Y^2\cdots Y^t\), and by a union bound over \(O(1/\nu \eta )\) blocks it suffices to treat this prefix window. We must show

Conditioning on \(t\) and using \(\mathbb {E}|Y_0|=\mathbb {E}|Y_1|\),

By the balance property of \(c\) (analogue of Proposition 28), each long substring of \(c\) has at least a \(\gamma \)-fraction of ones, so for \(t\ge \tfrac 12\mathbb {E}t\) (whp),

Since \(f{\lt}\tfrac 12\) and \(\gamma {\gt}0\), choosing \(\kappa \) small makes this \(-C\, \mathbb {E}t\) for a constant \(C{\gt}0\), i.e. \(-\Omega (m)\). By concentration (variances are \(o(m^2)\), via Chebyshev’s inequality, Theorem 5), the probability vanishes as \(m\to \infty \).

We adapt the construction of Definition 24, changing only buffer identification: declare any contiguous block of \(\nu \eta m\) bits with a fraction of ones less than \(f+\kappa \) to be part of a buffer (implemented with a running window of size \(\nu \eta m\); a buffer starts when the fraction drops below \(f+\kappa \) and ends when it rises above). The rate is as in Theorem 23 except \(n=km/(\mathrm{Cap}(\mathrm{DC})-\nu \eta -\psi (\varepsilon ,\delta ,\nu ,k,m))\) with \(\psi \to 0\); using Lemma 32 (information rate of the trimming Dobrushin channel is \(\ge \mathrm{Cap}(\mathrm{DC})-2\nu \eta \)) and taking \(\varepsilon ,\delta ,\eta ,\nu \) small and \(m\) large makes \(R\ge \mathrm{Cap}(\mathrm{DC})-\varepsilon \).

The error analysis mirrors the proof of Theorem 23; two additional claims are needed: (1) the probability that any length-\(\nu \eta m\) substring of a received buffer has a fraction of ones higher than \(f+\kappa \) vanishes as \(m\to \infty \) (an easier version of Lemma 33, with \(\nu \) small enough to identify buffers whp), and (2) the probability that any length-\(\nu \eta m\) substring of a received codeword has a fraction of ones lower than \(f+\kappa \) vanishes (Lemma 33). As in Theorem 23 each error type’s frequency is a binomial variable, so by the Chernoff bound (Theorem 8) and a union bound (Theorem 6) the failure probability is \(e^{-\Omega (n)}\). Encoding is linear and buffer identification (running window) is \(O(n)\), so decoding is quasi-linear.