New bounds for matching vector families

Abhishek Bhowmick, Zeev Dvir, Shachar Lovett

Research output: Chapter in Book/Report/Conference proceedingConference contribution

8 Scopus citations

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.

Original languageEnglish (US)
Title of host publicationSTOC 2013 - Proceedings of the 2013 ACM Symposium on Theory of Computing
Pages823-832
Number of pages10
DOIs
StatePublished - 2013
Event45th Annual ACM Symposium on Theory of Computing, STOC 2013 - Palo Alto, CA, United States
Duration: Jun 1 2013Jun 4 2013

Publication series

NameProceedings of the Annual ACM Symposium on Theory of Computing
ISSN (Print)0737-8017

Other

Other45th Annual ACM Symposium on Theory of Computing, STOC 2013
CountryUnited States
CityPalo Alto, CA
Period6/1/136/4/13

All Science Journal Classification (ASJC) codes

  • Software

Keywords

  • Locally decodable codes
  • Matching vector families
  • Restricted modular intersections

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  • Cite this

    Bhowmick, A., Dvir, Z., & Lovett, S. (2013). New bounds for matching vector families. In STOC 2013 - Proceedings of the 2013 ACM Symposium on Theory of Computing (pp. 823-832). (Proceedings of the Annual ACM Symposium on Theory of Computing). https://doi.org/10.1145/2488608.2488713