Matching-Vector Families and LDCs over Large Modulo

Abstract

We prove new upper bounds on the size of families of vectors in Z n m with restricted modular inner products, when m is a large integer. More formally, if u 1 , . . . , u t ∈ Z n m and v 1 , . . . , 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(

2-Server PIR with Sub-Polynomial Communication

“…Clearly, a construction of MV families in Z k 6 of larger size will immediately give better 2-server PIR schemes. There is very little known about the limitations of this approach and current upper bounds on the size of MV families are exponential and consistent with obtaining poly-logarithmic communication cost for 2-server protocols using our approach [7,4,8].…”

On the Inner Product Predicate and a Generalization of Matching Vector Families

“…If q is a product of k primes, we have a 2 Õ((log n) 1/k ) upper bound from [Gro00]. For any composite q, we also have an Ω(log n) lower bound from [DH13].…”