We show that any q -query locally decodable code (LDC) gives a copy of 1 k with small distortion in the Banach space of q -linear forms on N p 1 N p q , provided 1 p 1 + + 1 p q 1 and where k , N , and the distortion are simple functions of the code parameters. We exhibit the copies of 1 k by constructing a basis directly from "smooth" LDC decoders, thus bypassing a matching lemma often used in the LDC literature. Based on this, we give alternative proofs for known lower bounds on the length of 2-query LDCs. Using similar techniques, we reprove known lower bounds for larger q . We also discuss the relation with an alternative proof, due to Pisier, of a result of Naor, Regev, and the author on cotype properties of projective tensor products of p spaces.