A $(k,δ,ε)$-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 $d(\vec{y},C(\vec{x})) \leq δN$ and each $1 \leq i \leq n$, the symbol $x_{i}$ of $\vec{x}$ can be recovered with probability at least $1-ε$ by a randomized decoding algorithm looking only at $k$ coordinates of $\vec{y}$. The efficiency of a $(k,δ,ε)$-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 $N=\exp(n^{O((\log \log n/ \log n)^{1-1/r})})$.
Exponential Lower Bound for 2-Query Locally Decodable Codes via a Quantum Argument
局部可解码代码将n位串x编码为m位码字C（x），以这样的方式，即可以通过仅查询该字的几位来从损坏的码字中恢复任何位x_i。我们使用量子论证来证明具有2个经典查询的LDCs需要指数长度：m=2^{Omega（n）}。以前，这只适用于线性码（Goldreich et al. 02）。我们的证明表明，一个2查询LDC可以解码只有1个量子查询，然后证明了指数下界，这样的1查询局部量子可解码的代码。我们还表明，q量子查询允许更简洁的最不发达国家比最有名的最不发达国家与q经典查询。最后，我们给出了新的经典下界和量子上界的设置私人信息检索。特别是，我们展示了一个量子2-服务器PIR方案，具有O（n^{3/10}）量子比特的通信，改进了最著名的经典2-服务器PIR的O（n^{1/3}）比特的通信。
Exponential Lower Bounds for 2-query Relaxed Locally Decodable Codes
University of Illinois at Chicago（芝加哥伊利诺伊大学）；Purdue University（普渡大学）；Peking University（北京大学）；University of Waterloo（滑铁卢大学）；Johns Hopkins University（约翰斯·霍普金斯大学）
3-Query RLDCs are Strictly Stronger than 3-Query LDCs
Department of Computer Science and Technology, University of Cambridge（剑桥大学计算机科学与技术系）；Department of Mathematics, Massachusetts Institute of Technology（麻省理工学院数学系）；EPFL（洛桑联邦理工学院）
Exponential Lower Bounds for Locally Decodable and Correctable Codes for Insertions and Deletions
Purdue University（普渡大学）；Center on Frontiers of Computing Studies, Peking University（北京大学前沿计算研究中心）；Advanced Institute of Information Technology, Peking University（北京大学信息技术高级研究院）；Cheriton School of Computer Science, University of Waterloo（滑铁卢大学切里顿计算机科学学院）；Department of Computer Science, Johns Hopkins University（约翰斯·霍普金斯大学计算机科学系）
