On Embeddings of $\ell_1^k$ from Locally Decodable Codes
نویسنده
چکیده
We show that any q-query locally decodable code (LDC) gives a copy of l 1 with small distortion in the Banach space of q-linear forms on lNp1 ×· · ·× lNpq , provided 1/p1+ · · ·+1/pq ≤ 1 and where k, N , and the distortion are simple functions of the code parameters. We exhibit the copy of l 1 by constructing a basis for it directly from “smooth” LDC decoders. 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 lp spaces.
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