19 research outputs found
Local Decoding and Testing of Polynomials over Grids
The well-known DeMillo-Lipton-Schwartz-Zippel lemma says that n-variate polynomials of total degree at most d over grids, i.e. sets of the form A_1 times A_2 times cdots times A_n, form error-correcting codes (of distance at least 2^{-d} provided min_i{|A_i|}geq 2). In this work we explore their local decodability and local testability. While these aspects have been studied extensively when A_1 = cdots = A_n = F_q are the same finite field, the setting when A_i\u27s are not the full field does not seem to have been explored before.
In this work we focus on the case A_i = {0,1} for every i. We show that for every field (finite or otherwise) there is a test whose query complexity depends only on the degree (and not on the number of variables). In contrast we show that decodability is possible over fields of positive characteristic (with query complexity growing with the degree of the polynomial and the characteristic), but not over the reals, where the query complexity must grow with . As a consequence we get a natural example of a code (one with a transitive group of symmetries) that is locally testable but not locally decodable.
Classical results on local decoding and testing of polynomials have relied on the 2-transitive symmetries of the space of low-degree polynomials (under affine transformations). Grids do not possess this symmetry: So we introduce some new techniques to overcome this handicap and in particular use the hypercontractivity of the (constant weight) noise operator on the Hamming cube
A stability result for the cube edge isoperimetric inequality
We prove the following stability version of the edge isoperimetric inequality
for the cube: any subset of the cube with average boundary degree within of
the minimum possible is -close to a union of disjoint cubes,
where is independent of the dimension. This extends
a stability result of Ellis, and can viewed as a dimension-free version of
Friedgut's junta theorem.Comment: 12 page
Upper bound on list-decoding radius of binary codes
Consider the problem of packing Hamming balls of a given relative radius
subject to the constraint that they cover any point of the ambient Hamming
space with multiplicity at most . For odd an asymptotic upper bound
on the rate of any such packing is proven. Resulting bound improves the best
known bound (due to Blinovsky'1986) for rates below a certain threshold. Method
is a superposition of the linear-programming idea of Ashikhmin, Barg and Litsyn
(that was used previously to improve the estimates of Blinovsky for ) and
a Ramsey-theoretic technique of Blinovsky. As an application it is shown that
for all odd the slope of the rate-radius tradeoff is zero at zero rate.Comment: IEEE Trans. Inform. Theory, accepte