New bounds for matching vector families

• Authors: Abhishek Bhowmick;Zeev Dvir;Shachar Lovett
• Venue: Proceedings of the forty-fifth annual ACM symposium on Theory of computing
• Year: 2013

A Matching Vector (MV) family modulo m is a pair of ordered lists U=(u1,...,ut) and V=(v1,...,vt) where ui,vj ∈ Zmn with the following inner product pattern: for any i, {ui,vi}=0, and for any i ≠ j, {ui,vj} ≠ 0. A MV family is called q-restricted if inner products {ui,vj} take at most q different values. Our interest in MV families stems from their recent application in the construction of sub-exponential locally decodable codes (LDCs). There, q-restricted MV families are used to construct LDCs with q queries, and there is special interest in the regime where q is constant. When m is a prime it is known that such constructions yield codes with exponential block length. However, for composite m the behaviour is dramatically different. A recent work by Efremenko [8] (based on an approach initiated by Yekhanin [24]) gives the first sub-exponential LDC with constant queries. It is based on a construction of a MV family of super-polynomial size by Grolmusz [10] modulo composite m. In this work, we prove two lower bounds on the block length of LDCs which are based on black box construction using MV families. When q is constant (or sufficiently small), we prove that such LDCs must have a quadratic block length. When the modulus m is constant (as it is in the construction of Efremenko [8]) we prove a super-polynomial lower bound on the block-length of the LDCs, assuming a well-known conjecture in additive combinatorics, the polynomial Freiman-Ruzsa conjecture over Zm.