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

Document exchange and error correcting codes are two fundamental problems regarding communications. In both problems, an upper bound is placed on the number of errors between the two strings or that the channel can add, and a major goal is to minimize the size of the sketch or the redundant information. In this paper we focus on deterministic document exchange protocols and binary error correcting codes. 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. 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 $O(k \log^2 n+t)$ and explicit binary error correcting code with $O(k \log n \log \log n+t)$ redundant bits. As a comparison, the information-theoretic optimum for both problems is $\Theta(k \log n+t)$. As far as we know, previously there are no known explicit deterministic document exchange protocols in this case, and the best known binary code needs $\Omega(n)$ redundant bits even to correct just \emph{one} block transposition.