1 research outputs found
New Lower Bounds for Matching Vector Codes
A Matching Vector (MV) family modulo is a pair of ordered lists
and where with
the following inner product pattern: for any , , and for any
, . A MV family is called -restricted if inner
products take at most different values.
Our interest in MV families stems from their recent application in the
construction of sub-exponential locally decodable codes (LDCs). There,
-restricted MV families are used to construct LDCs with queries, and
there is special interest in the regime where is constant. When is a
prime it is known that such constructions yield codes with exponential block
length. However, for composite the behaviour is dramatically different. A
recent work by Efremenko [STOC 2009] (based on an approach initiated by
Yekhanin [JACM 2008]) 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 [Combinatorica 2000] modulo composite .
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 is constant (or
sufficiently small), we prove that such LDCs must have a quadratic block
length. When the modulus is constant (as it is in the construction of
Efremenko) 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 .Comment: Fixed typos and small bug