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We define the notion of a transitive-closure spanner of a directed graph. Given a directed graph G = (V, E) and an integer k ≥ 1, a k-transitive-closure-spanner (k-TC-spanner) of G is a directed graph H = (V, E H) that has (1) the same transitive-closure as G and (2) diameter at most k. These spanners were studied implicitly in access control, property… (More)

In the history of property testing, a particularly important role has been played by linear-invariant properties, i.e., properties of Boolean functions on the hypercube which are closed under linear transformations of the domain. Examples of such properties include linearity, Reed-Muller codes, and Fourier sparsity. In this work, we describe a framework… (More)

We introduce a framework for proving lower bounds on computational problems over distributions, based on a class of algorithms called <i>statistical algorithms</i>. For such algorithms, access to the input distribution is limited to obtaining an estimate of the expectation of any given function on a sample drawn randomly from the input distribution, rather… (More)

We consider the problem of learning sparse parities in the presence of noise. For learning parities on r out of n variables, we give an algorithm that runs in time poly log 1 δ , 1 1−2η n (1+(2η) 2 +o(1))r/2 and uses only r log(n/δ)ω(1) (1−2η) 2 samples in the random noise setting under the uniform distribution, where η is the noise rate and δ is the… (More)

We exhibit explicit bases for BCH codes of designed distance 5. While BCH codes are some of the most studied families of codes, only recently Kaufman and Litsyn (FOCS, 2005) showed that they admit bases of small weight codewords. Furthermore, Grigorescu, Kaufman and Sudan (RANDOM, 2009) and Kaufman and Lovett (FOCS, 2011) proved that in fact BCH codes can… (More)

Motivated by questions in property testing, we search for linear error-correcting codes that have the " single local orbit " property: i.e., they are specified by a single local constraint and its translations under the symmetry group of the code. We show that the dual of every " sparse " binary code whose coordinates are indexed by elements of F 2 n for… (More)

A basic goal in property testing is to identify a minimal set of features that make a property testable. For the case when the property to be tested is membership in a binary linear error-correcting code, Alon et al. (Trans Inf Theory, 51(11):4032–4039, 2005) had conjectured that the presence of a single low-weight codeword in the dual, and “2-transitivity”… (More)

Affine-invariant properties are an abstract class of properties that generalize some central algebraic ones, such as linearity and low-degree-ness, that have been studied extensively in the context of property testing. Affine invariant properties consider functions mapping a big field F q n to the subfield F q and include all properties that form an F… (More)

We develop a framework for proving lower bounds on computational problems over distributions , including optimization and unsupervised learning. Our framework is based on defining a restricted class of algorithms, called statistical algorithms, that instead of accessing samples from the input distribution can only obtain an estimate of the expectation of… (More)

Given a pair of finite groups G and H, the set of homomorphisms from G to H form an error-correcting code where codewords differ in at least 1/2 the coordinates. We show that for every pair of <i>abelian</i> groups G and H, the resulting code is (locally) list-decodable from a fraction of errors arbitrarily close to its distance. At the heart of this result… (More)