Locally decodable codes were introduced by Katz and Trevisan in 2000.
These are error correcting codes with the extra property that, in order 
to retrieve the correct value of just one position of the input with 
high probability,  it is 
sufficient to read a constant number of positions of the corresponding, 
possibly 
corrupted codeword.
A breakthrough result by Yekhanin showed that 3-query linear
locally decodable  codes may have subexponential length.

The construction of Yekhanin, and the  constructions that followed,
can achieve correctness only up to a certain limit which is
$1 - 3 \delta$ for nonbinary codes. 
The largest correctness for a subexponential size constant query binary 
code is achieved in a construction by Woodruff, and it is below 1 - 3 $\delta$,
where an adversary is allowed to corrupt up to $\delta$ fraction of 
the codeword. 

We show that achieving somewhat larger correctness (as a function of $\delta$)
requires exponential codeword length for 3-query linear codes.
Previously, there were no larger than quadratic lower bounds known for 
locally decodable codes with more than 2 queries.
Our lower bounds hold for linear codes over arbitrary finite fields.