Codes correcting deletions in oblivious and random models

We prove the existence of binary codes of positive rate that can correct an arbitrary pattern of $p$ fraction of deletions, for any $p 0$ and a code family with a randomized encoder $\mathsf{Enc}: \{0,1\}^{\mathcal{R} n} \to \{0,1\}^n$ and (deterministic) decoder $\mathsf{Dec}: \{0,1\}^{(1-p)n} \to \{0,1\}^{\mathcal{R} n}$ such that for all deletion patterns $\tau$ with $pn$ deletions and all messages $m \in \{0,1\}^{\mathcal{R} n}$, ${\textbf{Pr}} [ \mathsf{Dec}(\tau(\mathsf{Enc}(m))) \neq m ] \le o(1)$, where the probability is over the randomness of the encoder (which is private to the encoder and not known to the decoder). The oblivious model is a significant generalization of the random deletion channel where each bit is deleted independently with probability $p$. For the random deletion channel, existence of codes of rate $(1-p)/9$, and thus bounded away from $0$ for any $p 0$.