A $k$-query locally decodable code (LDC)
$\textbf{C}:\Sigma^{n}\rightarrow \Gamma^{N}$ encodes each message $x$ into
a codeword $\textbf{C}(x)$ such that each symbol of $x$ can be probabilistically
recovered by querying only $k$ coordinates of $\textbf{C}(x)$, even after a
constant fraction of the coordinates have been corrupted.
Yekhanin (2008)
constructed a $3$-query LDC of subexponential length,
$N=\exp(\exp (O(\log n/\log\log n)))$, under the assumption that there are
infinitely many Mersenne primes. Efremenko (2009) constructed a $3$-query LDC
of length $N_{2}=\exp(\exp (O(\sqrt{\log n\log\log n})))$ with no assumption, and a
$2^r$-query LDC of length $N_{r}=\exp(\exp(O(\sqrt[r]{\log n(\log \log n)^{r-1}})))$,
for every integer $r\geq 2$. Itoh and Suzuki (2010) gave a composition method in
Efremenko's framework and constructed a $3 \cdot 2^{r-2}$-query LDC of length
$N_{r}$, for every integer $r\geq 4$, which improved the query complexity of
Efremenko's LDC of the same length by a factor of $3/4$.
The main ingredient of
Efremenko's construction is the Grolmusz construction for super-polynomial
size set-systems with restricted intersections, over $\mathbb{Z}_m$, where
$m$ possesses a certain ``good'' algebraic property (related to the
``algebraic niceness'' property of Yekhanin (2008)).
Efremenko constructed a 3-query LDC based on $m=511$
and left as an open problem to find other
numbers that offer the same property for LDC constructions.

In this paper, we develop the algebraic theory behind the constructions of
Yekhanin (2008) and Efremenko (2009), in an attempt to understand
the ``algebraic niceness'' phenomenon in $\mathbb{Z}_m$.
We show that every integer
$m = pq = 2^t -1$, where $p$, $q$ and $t$ are prime, possesses the same
good algebraic property as $m=511$ that allows savings in query complexity.
We identify 50 numbers of this form by
computer search, which together with 511, are then applied to gain
improvements on query complexity via Itoh and Suzuki's composition method.
More precisely,
we construct a $3^{\lceil r/2\rceil}$-query LDC for every positive integer
$r<104$ and a $\left\lfloor (3/4)^{51}\cdot 2^{r}\right\rfloor$-query LDC
for every integer $r\geq 104$,
both of length $N_{r}$, improving the $2^r$ queries used by Efremenko (2009) and
$3\cdot 2^{r-2}$ queries used by Itoh and Suzuki (2010).

We also obtain new efficient private information retrieval (PIR)
schemes from the new query-efficient LDCs.