Skip to main content

New bounds for matching vector families

Author(s): Bhowmick, A; Dvir, Zeev; Lovett, S

To refer to this page use:
Abstract: A Matching Vector (MV) family modulo m is a pair of ordered lists U = (u1,..., ut) and V = (v1,...,vt) where ui, vj ∈ ℤn m with the following inner product pattern: for any i, (ui, vi)i = 0, and for any i ≠ j, (ui, vi)≠ 0. A MV family is called q-restricted if inner products hui; vji 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 rst 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.
Publication Date: 1-Jun-2013
Electronic Publication Date: 2013
Citation: Bhowmick, A, Dvir, Z, Lovett, S. (2013). New bounds for matching vector families. 823 - 832. doi:10.1145/2488608.2488713
DOI: doi:10.1145/2488608.2488713
Pages: 823 - 832
Type of Material: Conference Article
Journal/Proceeding Title: 45th Annual ACM Symposium on Theory of Computing, STOC 2013
Version: Author's manuscript

Items in OAR@Princeton are protected by copyright, with all rights reserved, unless otherwise indicated.