A Quadratic Lower Bound for Three-Query Linear Locally Decodable Codes over Any Field

A linear (q, δ, ϵ,m(n))-locally decodable code (LDC) C : $ {\mathbb F} $n→$ {\mathbb F} $m(n) is a linear transformation from the vector space $ {\mathbb F} $n to the space $ {\mathbb F} $m(n) for which each message symbol xi can be recovered with probability at least $ \frac{1}{{\left| \mathbb{F} \right|}} + \varepsilon $ from C(x) by a randomized algorithm that queries only q positions of C(x), even if up to δm(n) positions of C(x) are corrupted. In a recent work of Dvir, the author shows that lower bounds for linear LDCs can imply lower bounds for arithmetic circuits. He suggests that proving lower bounds for LDCs over the complex or real field is a good starting point for approaching one of his conjectures. Our main result is an m(n) = Ω(n2) lower bound for linear 3-query LDCs over any, possibly infinite, field. The constant in the Ω(·) depends only on ε and δ. This is the first lower bound better than the trivial m(n) = Ω(n) for arbitrary fields and more than two queries.