Improved Constructions for Query-Efficient Locally Decodable Codes of Subexponential Length

抄録

A (k, δ, ε)-locally decodable code $C:{\bf F}_{q}^{n} \ ightarrow {\bf F}_{q}^{N}$ is an error-correcting code that encodes $\vec{x}=(x_{1},x_{2},\ldots,x_{n}) \in {\bf F}_{q}^{n}$ to $C(\vec{x}) \in {\bf F}_{q}^{N}$ 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 ≤ i ≤ n, the symbol x_i of $ of $\vec{x}$ can be recovered with probability at least 1 - ε by a randomized decoding algorithm looking at only k coordinates of $\vec{y}$. The efficiency of a (k, δ, ε)-locally decodable code $C:{\bf F}_{q}^{n} \ ightarrow {\bf F}_{q}^{N}$ is measured by the code length N and the number k of queries. For a k-query locally decodable code $C:{\bf F}_{q}^{n} \ ightarrow {\bf F}_{q}^{N}$, the code length N was conjectured to be exponential of n, i.e., N = exp(n^Ω(1)), however, this was disproved. Yekhanin [In Proc. of STOC, 2007] showed that there exists a 3-query locally decodable code $C:{\bf F}_{2}^{n} \ ightarrow {\bf F}_{2}^{N}$ such that N=exp(n^{1/log log n}) assuming that infinitely many Mersenne primes exist. For a 3-query locally decodable code $C:{\bf F}_{q}^{n} \ ightarrow {\bf F}_{q}^{N}$, Efremenko [ECCC Report No.69, 2008] further reduced the code length to $N=\exp(n^{O((\log \log n/ \log n)^{1/2})})$, and in general showed that for any integer r > 1, there exists a 2^r-query locally decodable code $C:{\bf F}_{q}^{n} \ ightarrow {\bf F}_{q}^{N}$ such that $N=\exp(n^{O((\log \log n/ \log n)^{1-1/r})})$. In this paper, we will present improved constructions for query-efficient locally decodable codes by introducing a technique of “composition of locally decodable codes,” and show that for any integer r > 5, there exists a 9 · 2^r-4-query locally decodable code $C:{\bf F}_{q}^{n} \ ightarrow {\bf F}_{q}^{N}$ such that $N=\exp(n^{O((\log \log n/ \log n)^{1-1/r})})$.