On Embeddings of l1k from Locally Decodable Codes

We show that any q-query locally decodable code (LDC) gives a copy of `1 with small distortion in the Banach space of q-linear forms on `Np1×· · ·× ` N pq , provided 1/p1 + · · ·+1/pq ≤ 1 and where k, N , and the distortion are simple functions of the code parameters. We exhibit the copies of `1 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.