Matching-vector families and LDCs over large modulo

We prove new upper bounds on the size of families of vectors in ℤ_mⁿ with restricted modular inner products, when m is a large integer. More formally, if u_i,...,u_t ∈ ℤ_mⁿ and v₁,...,v_t ∈ ℤ_mⁿ satisfy 〈u_i, v_i〉 ≡ 0 (mod m) and 〈u_i, v_j〉 ≢ 0 (mod m) for all i ≠ j ∈ [t], we prove that t ≤ O(m^n/2+8.47). This improves a recent bound of t ≤ m^{n/2+O(log(m))} by [BDL13] and is the best possible up to the constant 8.47 when m is sufficiently larger than n. The maximal size of such families, called 'Matching-Vector families', shows up in recent constructions of locally decodable error correcting codes (LDCs) and determines the rate of the code. Using our result we are able to show that these codes, called Matching-Vector codes, must have encoding length at least K^19/18 for K-bit messages, regardless of their query complexity. This improves a known super linear bound of K2^{Ω(√log K)} proved in [BDL13].