  We study two fundamental problems in communication, Document Exchange (DE)
and Error Correcting Code (ECC). In the first problem, two parties hold two
strings, and one party tries to learn the other party's string through
communication. In the second problem, one party tries to send a message to
another party through a noisy channel, by adding some redundant information to
protect the message. Two important goals in both problems are to minimize the
communication complexity or redundancy, and to design efficient protocols or
codes.
  Both problems have been studied extensively. In this paper we study whether
asymmetric partial information can help in these two problems. We focus on the
case of Hamming distance/errors, and the asymmetric partial information is
modeled by one party having a vector of disjoint subsets $\vec{S}=(S_1, \cdots,
S_t)$ of indices and a vector of integers $\vec{k}=(k_1, \cdots, k_t)$, such
that in each $S_i$ the Hamming distance/errors is at most $k_i$. We establish
both lower bounds and upper bounds in this model, and provide efficient
randomized constructions that achieve a $\min\lbrace O(t^2), O\left((\log \log
n)^2\right) \rbrace $ factor within the optimum, with almost linear running
time.
  We further show a connection between the above document exchange problem and
the problem of document exchange under edit distance, and use our techniques to
give an efficient randomized protocol with optimal communication complexity and
\emph{exponentially} small error for the latter. This improves the previous
result by Haeupler \cite{haeupler2018optimal} (FOCS'19) and that by Belazzougui
and Zhang \cite{BelazzouguiZ16} (FOCS'16). Our techniques are based on a
generalization of the celebrated expander codes by Sipser and Spielman
\cite{sipser1996expander}, which may be of independent interests.
