Matching-vector families and LDCs over large modulo

Abstract

We prove new upper bounds on the size of families of vectors in ℤm n with restricted modular inner products, when m is a large integer. More formally, if ui,...,ut ∈ ℤm n and v1,...,vt ∈ ℤm n satisfy 〈ui, vi〉 ≡ 0 (mod m) and 〈ui, vj〉 ≢ 0 (mod m) for all i ≠ j ∈ [t], we prove that t ≤ O(mn/2+8.47). This improves a recent bound of t ≤ mn/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 K19/18 for K-bit messages, regardless of their query complexity. This improves a known super linear bound of K2Ω(√log K) proved in [BDL13]