Locally decodable codes and the failure of cotype for projective tensor products

Abstract : It is shown that for every p is an element of (1, infinity) there exists a Banach space X of finite cotype such that the projective tensor product l(p) (circle times) over cap X fails to have finite cotype. More generally, if p(1); p(2); p(3) is an element of (1, infinity) satisfy 1/p(1) + 1/p(2) + 1/p(3) <= 1 then l(p1) (circle times) over capl(p2)(circle times) over capl(p3) does not have fi nite cotype. This is proved via a connection to the theory of locally decodable codes.