3-Query Locally Decodable Codes of Subexponential Length

Locally Decodable Codes (LDC) allow one to decode any particular symbol of the input message by making a constant number of queries to a codeword, even if a constant fraction of the codeword is damaged. In a recent work [Yek08] Yekhanin constructs a 3-query LDC with sub-exponential length of size exp(exp(O((log n)/(log log n)))). However, this construction requires a conjecture that there are infinitely many Mersenne primes. In this paper we give the first unconditional constant query LDC construction with subexponantial codeword length. In addition our construction reduces codeword length. We give construction of 3-query LDC with codeword length exp(exp(O(√{log n log log n ))). Our construction could also be extended to higher number of queries. We give a 2r-query LDC with length of exp(exp(O(☂[r] log n (log log n)r-1))).

[1]  Sergey Yekhanin,et al.  Locally Decodable Codes from Nice Subsets of Finite Fields and Prime Factors of Mersenne Numbers , 2009, SIAM J. Comput..

[2]  Yuval Ishai,et al.  Breaking the O(n/sup 1/(2k-1)/) barrier for information-theoretic Private Information Retrieval , 2002, The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings..

[3]  Yasuhiro Suzuki,et al.  Improved Constructions for Query-Efficient Locally Decodable Codes of Subexponential Length , 2008, IEICE Trans. Inf. Syst..

[4]  Prasad Raghavendra,et al.  A Note on Yekhanin's Locally Decodable Codes , 2007, Electron. Colloquium Comput. Complex..

[5]  Ronald de Wolf,et al.  Improved Lower Bounds for Locally Decodable Codes and Private Information Retrieval , 2004, ICALP.

[6]  David P. Woodruff.  New Lower Bounds for General Locally Decodable Codes , 2007, Electron. Colloquium Comput. Complex..

[7]  Ashwin Nayak,et al.  Optimal lower bounds for quantum automata and random access codes , 1999, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

[8]  Luca Trevisan,et al.  Some Applications of Coding Theory in Computational Complexity , 2004, Electron. Colloquium Comput. Complex..

[9]  Ronald de Wolf,et al.  Exponential lower bound for 2-query locally decodable codes via a quantum argument , 2002, STOC '03.

[10]  Toshiya Itoh,et al.  Efficient Private Information Retrieval , 1999 .

[11]  David P. Woodruff.  Corruption and Recovery-Efficient Locally Decodable Codes , 2008, APPROX-RANDOM.

[12]  Luca Trevisan,et al.  Lower bounds for linear locally decodable codes and private information retrieval , 2002, Proceedings 17th IEEE Annual Conference on Computational Complexity.

[13]  Thierry Paul,et al.  Quantum computation and quantum information , 2007, Mathematical Structures in Computer Science.

[14]  Andris Ambainis,et al.  Dense quantum coding and a lower bound for 1-way quantum automata , 1998, STOC '99.

[15]  Andris Ambainis,et al.  On Lower Bounds for the Communication Complexity of Private Information Retrieval ∗ , 2000 .

[16]  William I. Gasarch,et al.  A Survey on Private Information Retrieval (Column: Computational Complexity) , 2004, Bull. EATCS.

[17]  Sergey Yekhanin.  Towards 3-query locally decodable codes of subexponential length , 2007, STOC '07.

[18]  Iordanis Kerenidis.  Quantum multiparty communication complexity and circuit lower bounds , 2009, Math. Struct. Comput. Sci..

[19]  Vince Grolmusz,et al.  Superpolynomial Size Set-systems with Restricted Intersections mod 6 and Explicit Ramsey Graphs , 2000, Comb..

[20]  Jonathan Katz,et al.  On the efficiency of local decoding procedures for error-correcting codes , 2000, STOC '00.

[21]  Alexander A. Razborov,et al.  An Omega(n1/3) Lower Bound for Bilinear Group Based Private Information Retrieval , 2007, Theory of Computing.

[22]  Tao Feng,et al.  Query-Efficient Locally Decodable Codes of Subexponential Length , 2010, computational complexity.

[23]  William Gasarch.  A Survey on Private Information Retrieval , 2004 .

[24]  Leonid A. Levin,et al.  Checking computations in polylogarithmic time , 1991, STOC '91.

[25]  Yuval Ishai,et al.  General constructions for information-theoretic private information retrieval , 2005, J. Comput. Syst. Sci..

[26]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[27]  Eyal Kushilevitz,et al.  Private information retrieval , 1995, Proceedings of IEEE 36th Annual Foundations of Computer Science.

[28]  Elizabeth D Mann.  Private access to distributed information , 1998 .

[29]  Yuval Ishai,et al.  Information-Theoretic Private Information Retrieval: A Unified Construction , 2001, ICALP.