Some optimal inapproximability results
STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
Recycling queries in PCPs and in linearity tests (extended abstract)
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
The approximability of NP-hard problems
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
Proof verification and the hardness of approximation problems
Journal of the ACM (JACM)
Some optimal inapproximability results
Journal of the ACM (JACM)
Simple analysis of graph tests for linearity and PCP
Random Structures & Algorithms
Property testing in massive graphs
Handbook of massive data sets
Randomness-efficient low degree tests and short PCPs via epsilon-biased sets
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
Algebraic testing and weight distributions of codes
Theoretical Computer Science
Derandomizing homomorphism testing in general groups
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Almost Orthogonal Linear Codes are Locally Testable
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
Gowers uniformity, influence of variables, and PCPs
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Robust locally testable codes and products of codes
Random Structures & Algorithms
Locally testable codes and PCPs of almost-linear length
Journal of the ACM (JACM)
Low-degree tests at large distances
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Non-Abelian homomorphism testing, and distributions close to their self-convolutions
Random Structures & Algorithms
Inverse conjecture for the gowers norm is false
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Unconditional pseudorandom generators for low degree polynomials
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Property Testing: A Learning Theory Perspective
Foundations and Trends® in Machine Learning
On proximity oblivious testing
Proceedings of the forty-first annual ACM symposium on Theory of computing
Testing Fourier Dimensionality and Sparsity
ICALP '09 Proceedings of the 36th International Colloquium on Automata, Languages and Programming: Part I
ACM Transactions on Computation Theory (TOCT)
Algorithmic and Analysis Techniques in Property Testing
Foundations and Trends® in Theoretical Computer Science
Computing partial Walsh transform from the algebraic normal form of a Boolean function
IEEE Transactions on Information Theory
Local list-decoding and testing of random linear codes from high error
Proceedings of the forty-second ACM symposium on Theory of computing
A query efficient non-adaptive long code test with perfect completeness
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
Breaking the $\epsilon$-Soundness Bound of the Linearity Test over GF(2)
SIAM Journal on Computing
Guest column: testing linear properties: some general theme
ACM SIGACT News
Limitation on the rate of families of locally testable codes
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Testing juntas: a brief survey
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Limitation on the rate of families of locally testable codes
Property testing
Testing juntas: a brief survey
Property testing
Optimal Testing of Reed-Muller Code
Property testing
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Property testing
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Proceedings of the forty-third annual ACM symposium on Theory of computing
Characterizations of locally testable linear-and affine-invariant families
COCOON'11 Proceedings of the 17th annual international conference on Computing and combinatorics
Characterizations of locally testable linear- and affine-invariant families
Theoretical Computer Science
On Proximity-Oblivious Testing
SIAM Journal on Computing
Testing Fourier Dimensionality and Sparsity
SIAM Journal on Computing
On the distance between non-isomorphic groups
European Journal of Combinatorics
Local decoding and testing for homomorphisms
APPROX'06/RANDOM'06 Proceedings of the 9th international conference on Approximation Algorithms for Combinatorial Optimization Problems, and 10th international conference on Randomization and Computation
SIAM Journal on Discrete Mathematics
Taking proof-based verified computation a few steps closer to practicality
Security'12 Proceedings of the 21st USENIX conference on Security symposium
On the structure of boolean functions with small spectral norm
Proceedings of the 5th conference on Innovations in theoretical computer science
High dimensional expanders and property testing
Proceedings of the 5th conference on Innovations in theoretical computer science
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Let Dist(f,g)=Pru[f(u)≠g(u)] denote the relative distance between functions f,g mapping from a group G to a group H, and let Dist(f) denote the minimum, over all linear functions (homomorphisms) g, of Dist(f,g). Given a function f:G→H we let Err(f)=Pru,υ[f(u)+f(υ)≠f(u+υ)] denote the rejection probability of the Blum-Luby-Rubinfeld (1993) linearity test. Linearity testing is the study of the relationship between Err(f) and Dist(f), and in particular lower bounds on Err(f) in terms of Dist(f). We discuss when the underlying groups are G=GF(2)n and H=GF(2). In this case, the collection of linear functions describe a Hadamard code of block length 2n and for an arbitrary function f mapping GF(2)n to GF(2) the distance Dist(l) measures its distance to a Hadamard code. Err(f) is a parameter that is “easy to measure” and linearity testing studies the relationship of this parameter to the distance of f. The code and corresponding test are used in the construction of efficient probabilistically checkable proofs and thence in the derivation of hardness of approximation. Improved analyses translate into better nonapproximability results. We present a description of the relationship between Err(f) and Dist(f) which is nearly complete in all its aspects, and entirely complete in some. We present functions L,U:[0,1]→[0,1] such that for all x ∈ [0,1] we have L(x)⩽Err(f)⩽U(x) whenever Dist(f)=x, with the upper bound being tight on the whole range, and the lower bound tight on a large part of the range and close on the rest. Part of our strengthening is obtained by showing a new connection between linearity testing and Fourier analysis