Tali Kaufman

Testing Reed-Muller codes

A code is locally testable if there is a way to indicate with high probability that a vector is far enough from any codeword by accessing only a very small number of the vectors bits. We show that the Reed-Muller codes of constant order are locally testable. Specifically, we describe an efficient randomized algorithm to test if a given vector of length n=2/sup m/ is a word in the rth-order Reed-Muller code R(r,m) of length n=2/sup m/. For a given integer r/spl ges/1, and real /spl epsi/>0, the algorithm queries the input vector /spl upsi/ at O(1//spl epsi/+r2/sup 2r/) positions. On the one hand, if /spl upsi/ is at distance at least /spl epsi/n from the closest codeword, then the algorithm discovers it with probability at least 2/3. On the other hand, if /spl upsi/ is a codeword, then it always passes the test. Our result is almost tight: any algorithm for testing R(r,m) must perform /spl Omega/(1//spl epsi/+2/sup r/) queries.

Testing Low-Degree Polynomials over GF(2)

We describe an efficient randomized algorithm to test if a given binary function f: {0,1} n →{0,1} is a low-degree polynomial (that is, a sum of low-degree monomials). For a given integer k ≥ 1 and a given real e >0, the algorithm queries f at \(O(\frac{1}{\epsilon}+k4^k)\) points. If f is a polynomial of degree at most k, the algorithm always accepts, and if the value of f has to be modified on at least an e fraction of all inputs in order to transform it to such a polynomial, then the algorithm rejects with probability at least 2/3. Our result is essentially tight: Any algorithm for testing degree-k polynomials over GF(2) must perform \(\Omega(\frac{1}{\epsilon}+2^k)\) queries.

Testing k-wise and almost k-wise independence

In this work, we consider the problems of testing whether adistribution over (0,1ⁿ) is <i>k</i>-wise (resp. (ε,k)-wise) independentusing samples drawn from that distribution. For the problem of distinguishing <i>k</i>-wise independent distributions from those that are δ-far from <i>k</i>-wise independence in statistical distance, we upper bound the number ofrequired samples by Õ(n^k/δ²) and lower bound it by Ω(n^k-1/2/δ) (these bounds hold for constant<i>k</i>, and essentially the same bounds hold for general <i>k</i>). Toachieve these bounds, we use Fourier analysis to relate adistributions distance from <i>k</i>-wise independence to its biases, a measure of the parity imbalance it induces on a setof variables. The relationships we derive are tighter than previouslyknown, and may be of independent interest. To distinguish (ε,k)-wise independent distributions from thosethat are δ-far from (ε,k)-wise independence in statistical distance, we upper bound thenumber of required samples by O(k log n / δ²ε²) and lower bound it by Ω(√ k log n / 2^k(ε+δ)√ log 1/2^k(ε+δ)). Although these bounds are anexponential improvement (in terms of <i>n</i> and <i>k</i>) over thecorresponding bounds for testing <i>k</i>-wise independence, we give evidence thatthe time complexity of testing (ε,k)-wise independence isunlikely to be poly(n,1/ε,1/δ) for k=Θ(log n),since this would disprove a plausible conjecture concerning the hardness offinding hidden cliques in random graphs. Under the conjecture, ourresult implies that for, say, k = log n and ε = 1 / n^0.99,there is a set of (ε,k)-wise independent distributions, and a set of distributions at distance δ=1/n^0.51 from (ε,k)-wiseindependence, which are indistinguishable by polynomial time algorithms.

Tight Bounds for Testing Bipartiteness in General Graphs

In this paper we consider the problem of testing bipartiteness of general graphs. The problem has previously been studied in two models, one most suitable for dense graphs and one most suitable for bounded-degree graphs. Roughly speaking, dense graphs can be tested for bipartiteness with constant complexity, while the complexity of testing bounded-degree graphs is

Testing polynomials over general fields

\tilde{\Theta}(\sqrt{n})

Almost orthogonal linear codes are locally testable

, where

Worst Case to Average Case Reductions for Polynomials

n

Testing Polynomials over General Fields

is the number of vertices in the graph (and

Verifying and decoding in constant depth

\tilde{\Theta}(f(n))

Weight Distribution and List-Decoding Size of Reed–Muller Codes

means