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

Abstract

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