  A k-query Locally Decodable Code (LDC) encodes an n-bit message x as an N-bit
codeword C(x), such that one can probabilistically recover any bit x_i of the
message by querying only k bits of the codeword C(x), even after some constant
fraction of codeword bits has been corrupted. The major goal of LDC related
research is to establish the optimal trade-off between length and query
complexity of such codes.
  Recently [Y] introduced a novel technique for constructing locally decodable
codes and vastly improved the upper bounds for code length. The technique is
based on Mersenne primes. In this paper we extend the work of [Y] and argue
that further progress via these methods is tied to progress on an old number
theory question regarding the size of the largest prime factors of Mersenne
numbers.
  Specifically, we show that every Mersenne number m=2^t-1 that has a prime
factor p>m^\gamma yields a family of k(\gamma)-query locally decodable codes of
length Exp(n^{1/t}). Conversely, if for some fixed k and all \epsilon > 0 one
can use the technique of [Y] to obtain a family of k-query LDCs of length
Exp(n^\epsilon); then infinitely many Mersenne numbers have prime factors arger
than known currently.
