New Bounds For Matching Vector Families

A matching vector (MV) family modulo m is a pair of ordered lists U = (u(1),...,u(t)) and V = (v(1),...,v(t)) where u(i), v(j) is an element of Z(m)(n) with the following inner product pattern: for any i, < u(i), v(i)> = 0, and for any i not equal j, < u(i), v(j)> not equal 0. An MV family is called q-restricted if inner products < u(i), v(j)> take at most q different values. Our interest in MV families stems from their recent application in the construction of subexponential 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 behavior is dramatically different. A recent work by Efremenko [SIAM J. Comput., 40 (2011), pp. 1154-1178] (based on an approach initiated by Yekhanin [J. ACM, 55 (2008), pp. 1-16]) gives the first subexponential LDC with constant queries. It is based on a construction of an MV family of superpolynomial size by Grolmusz [Combinatorica, 20 (2000), pp. 71-86] 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) we prove a superpolynomial lower bound on the block-length of the LDCs, assuming a well-known conjecture in additive combinatorics, the polynomial Freiman-Ruzsa conjecture over Z(m).