Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: New Constructions for Query-Efficient Locally Decodable Codes of
  Subexponential Length   A $(k,\delta,\epsilon)$-locally decodable code $C: F_{q}^{n} \to F_{q}^{N}$
is an error-correcting code that encodes each message
$\vec{x}=(x_{1},x_{2},...,x_{n}) \in F_{q}^{n}$ to $C(\vec{x}) \in F_{q}^{N}$
and has the following property: For any $\vec{y} \in {\bf F}_{q}^{N}$ such that
$d(\vec{y},C(\vec{x})) \leq \delta N$ and each $1 \leq i \leq n$, the symbol
$x_{i}$ of $\vec{x}$ can be recovered with probability at least $1-\epsilon$ by
a randomized decoding algorithm looking only at $k$ coordinates of $\vec{y}$.
The efficiency of a $(k,\delta,\epsilon)$-locally decodable code $C: F_{q}^{n}
\to F_{q}^{N}$ is measured by the code length $N$ and the number $k$ of
queries. For any $k$-query locally decodable code $C: F_{q}^{n} \to F_{q}^{N}$,
the code length $N$ is conjectured to be exponential of $n$, however, this was
disproved. Yekhanin [In Proc. of STOC, 2007] showed that there exists a 3-query
locally decodable code $C: F_{2}^{n} \to F_{2}^{N}$ such that
$N=\exp(n^{(1/\log \log n)})$ assuming that the number of Mersenne primes is
infinite. For a 3-query locally decodable code $C: F_{q}^{n} \to F_{q}^{N}$,
Efremenko [ECCC Report No.69, 2008] reduced the code length further to
$N=\exp(n^{O((\log \log n/ \log n)^{1/2})})$, and also showed that for any
integer $r>1$, there exists a $k$-query locally decodable code $C: F_{q}^{n}
\to F_{q}^{N}$ such that $k \leq 2^{r}$ and $N=\exp(n^{O((\log \log n/ \log
n)^{1-1/r})})$. In this paper, we present a query-efficient locally decodable
code and show that for any integer $r>1$, there exists a $k$-query locally
decodable code $C: F_{q}^{n} \to F_{q}^{N}$ such that $k \leq 3 \cdot 2^{r-2}$
and $N=\exp(n^{O((\log \log n/ \log n)^{1-1/r})})$.

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The categories of this paper are: cmp-lg
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The categories of this paper are: adap-org
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The categories of this paper are: patt-sol
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The categories of this paper are: gr-qc
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The categories of this paper are: nlin
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The categories of this paper are: plasm-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: math
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The categories of this paper are: supr-con
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The categories of this paper are: nucl-th
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The categories of this paper are: hep-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: chao-dyn
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The categories of this paper are: q-bio
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The categories of this paper are: dg-ga
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The categories of this paper are: econ