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 Rd so that, for each i ∈ [d] there are Ω(δn) disjoint q-tuples (u1,...,uq) in V so that span(u1,...,uq) contains a unit vector whose i’th coordinate is at least α. We prove exponential lower bounds of the form n ≥ 2Ω(αδ d) for the case q = 2 and, in some cases, stronger bounds (exponential in d).