A (k,Ξ΄,Ο΅)-locally decodable code C:FqnββFqNβ
is an error-correcting code that encodes each message
x=(x1β,x2β,...,xnβ)βFqnβ to C(x)βFqNβ
and has the following property: For any yββFqNβ such that
d(yβ,C(x))β€Ξ΄N and each 1β€iβ€n, the symbol
xiβ of x can be recovered with probability at least 1βΟ΅ by
a randomized decoding algorithm looking only at k coordinates of yβ.
The efficiency of a (k,Ξ΄,Ο΅)-locally decodable code C:FqnββFqNβ is measured by the code length N and the number k of
queries. For any k-query locally decodable code C:FqnββFqNβ,
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:F2nββF2Nβ such that
N=exp(n(1/loglogn)) assuming that the number of Mersenne primes is
infinite. For a 3-query locally decodable code C:FqnββFqNβ,
Efremenko [ECCC Report No.69, 2008] reduced the code length further to
N=exp(nO((loglogn/logn)1/2)), and also showed that for any
integer r>1, there exists a k-query locally decodable code C:FqnββFqNβ such that kβ€2r and N=exp(nO((loglogn/logn)1β1/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:FqnββFqNβ such that kβ€3β 2rβ2
and N=exp(nO((loglogn/logn)1β1/r)).Comment: 13 pages, 1 figure, 2 table