{"paper":{"title":"On Embeddings of $\\ell_1^k$ from Locally Decodable Codes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CC","authors_text":"Jop Bri\\\"et","submitted_at":"2016-11-19T15:39:20Z","abstract_excerpt":"We show that any $q$-query locally decodable code (LDC) gives a copy of $\\ell_1^k$ with small distortion in the Banach space of $q$-linear forms on $\\ell_{p_1}^N\\times\\cdots\\times\\ell_{p_q}^N$, provided $1/p_1 + \\cdots + 1/p_q \\leq 1$ and where $k$, $N$, and the distortion are simple functions of the code parameters. We exhibit the copy of $\\ell_1^k$ 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 dis"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.06385","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}