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 to have 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 proved via a connection to the theory of locally decodable codes.

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American Institute of Mathematical Sciences
Electronic Research Announcements in Mathematical Sciences
Qubit Applications
Algorithms and Complexity

Briët, J, Naor, A, & Regev, O. (2012). Locally decodable codes and the failure of cotype for projective tensor products. Electronic Research Announcements in Mathematical Sciences, 19, 120–130.