Lower Bounds for Approximate LDCs

We study an approximate version of q-query LDCs (Locally Decodable Codes) over the real numbers and prove lower bounds on the encoding length of such codes. A q-query (α,δ)-approximate LDC is a set V of n points in ℝ d so that, for each i ∈ [d] there are Ω(δn) disjoint q-tuples (u 1,…,u q ) in V so that span(u 1,…,u q ) contains a unit vector whose i’th coordinate is at least α. We prove exponential lower bounds of the form \(n \geq 2^{\Omega(\alpha \delta \sqrt{d})}\) for the case q = 2 and, in some cases, stronger bounds (exponential in d).

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