  The \emph{longest common extension} (\emph{LCE}) problem is to preprocess a
given string $w$ of length $n$ so that the length of the longest common prefix
between suffixes of $w$ that start at any two given positions is answered
quickly. In this paper, we present a data structure of $O(z \tau^2 +
\frac{n}{\tau})$ words of space which answers LCE queries in $O(1)$ time and
can be built in $O(n \log \sigma)$ time, where $1 \leq \tau \leq \sqrt{n}$ is a
parameter, $z$ is the size of the Lempel-Ziv 77 factorization of $w$ and
$\sigma$ is the alphabet size. This is an \emph{encoding} data structure, i.e.,
it does not access the input string $w$ when answering queries and thus $w$ can
be deleted after preprocessing. On top of this main result, we obtain further
results using (variants of) our LCE data structure, which include the
following:
  - For highly repetitive strings where the $z\tau^2$ term is dominated by
$\frac{n}{\tau}$, we obtain a \emph{constant-time and sub-linear space} LCE
query data structure.
  - Even when the input string is not well compressible via Lempel-Ziv 77
factorization, we still can obtain a \emph{constant-time and sub-linear space}
LCE data structure for suitable $\tau$ and for $\sigma \leq 2^{o(\log n)}$.
  - The time-space trade-off lower bounds for the LCE problem by Bille et al.
[J. Discrete Algorithms, 25:42-50, 2014] and by Kosolobov [CoRR,
abs/1611.02891, 2016] can be "surpassed" in some cases with our LCE data
structure.
