It is shown that for every p∈(1,∞) there exists a Banach space X
of finite cotype such that the projective tensor product \ell_p\tp X fails to
have finite cotype. More generally, if p1,p2,p3∈(1,∞) satisfy
p11+p21+p31≤1 then
\ell_{p_1}\tp\ell_{p_2}\tp\ell_{p_3} does not have finite cotype. This is a
proved via a connection to the theory of locally decodable codes