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It is shown that for every $p\in (1,\infty)$ there exists a Banach space $X$of finite cotype such that the projective tensor product $\ell_p\tp X$ fails tohave finite cotype. More generally, if $p_1,p_2,p_3\in (1,\infty)$ satisfy$\frac{1}{p_1}+\frac{1}{p_2}+\frac{1}{p_3}\le 1$ then$\ell_{p_1}\tp\ell_{p_2}\tp\ell_{p_3}$ does not have finite cotype. This is aproved via a connection to the theory of locally decodable codes.

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