Block Edit Errors with Transpositions: Deterministic Document Exchange Protocols and Almost Optimal Binary Codes

Document exchange and error correcting codes are two fundamental problems regarding communications. In the first problem, Alice and Bob each holds a string, and the goal is for Alice to send a short sketch to Bob, so that Bob can recover Alice's string. In the second problem, Alice sends a message with some redundant information to Bob through a channel that can add adversarial errors, and the goal is for Bob to correctly recover the message despite the errors. In a recent work \cite{CJLW18}, the authors constructed explicit deterministic document exchange protocols and binary error correcting codes for edit errors with almost optimal parameters.\ Unfortunately, the constructions in \cite{CJLW18} do not work for other common errors such as block transpositions. In this paper, we generalize the constructions in \cite{CJLW18} to handle a much larger class of errors. These include bursts of insertions and deletions, as well as block transpositions. Specifically, we consider document exchange and error correcting codes where the total number of block insertions, block deletions, and block transpositions is at most $k \leq \alpha n/\log n$ for some constant $0<\alpha<1$. In addition, the total number of bits inserted and deleted by the first two kinds of operations is at most $t \leq \beta n$ for some constant $0<\beta<1$, where $n$ is the length of Alice's string or message. We construct explicit, deterministic document exchange protocols with sketch size $O( (k \log n +t) \log^2 \frac{n}{k\log n + t} )$ and explicit binary error correcting code with $O(k \log n \log \log \log n+t)$ redundant bits.