A hard-core predicate for all one-way functions
STOC '89 Proceedings of the twenty-first annual ACM symposium on Theory of computing
Learning decision trees using the Fourier spectrum
SIAM Journal on Computing
Self-testing/correcting with applications to numerical problems
Journal of Computer and System Sciences - Special issue: papers from the 22nd ACM symposium on the theory of computing, May 14–16, 1990
Pseudorandom generators without the XOR Lemma (extended abstract)
STOC '99 Proceedings of the thirty-first annual ACM symposium on Theory of computing
List decoding algorithms for certain concatenated codes
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
Learning Polynomials with Queries: The Highly Noisy Case
SIAM Journal on Discrete Mathematics
Exact and Approximate Testing/Correcting of Algebraic Functions: A Survey
Theoretical Aspects of Computer Science, Advanced Lectures [First Summer School on Theoretical Aspects of Computer Science, Tehran, Iran, July 2000]
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
Sub-constant error low degree test of almost-linear size
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Linearity testing in characteristic two
IEEE Transactions on Information Theory - Part 1
Decodability of group homomorphisms beyond the johnson bound
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
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Locally decodable codes (LDCs) have played a central role in many recent results in theoretical computer science. The role of finite fields, and in particular, low-degree polynomials over finite fields, in the construction of these objects is well studied. However the role of group homomorphisms in the construction of such codes is not as widely studied. Here we initiate a systematic study of local decoding of codes based on group homomorphisms. We give an efficient list decoder for the class of homomorphisms from any abelian group G to a fixed abelian group H. The running time of this algorithm is bounded by a polynomial in log|G| and an agreement parameter, where the degree of the polynomial depends on H. Central to this algorithmic result is a combinatorial result bounding the number of homomorphisms that have large agreement with any function from G to H. Our results give a new generalization of the classical work of Goldreich and Levin, and give new abstractions of the list decoder of Sudan, Trevisan and Vadhan. As a by-product we also derive a simple(r) proof of the local testability (beyond the Blum-Luby-Rubinfeld bounds) of homomorphisms mapping ${\mathbb{Z}}_p^n$ to ℤp, first shown by M. Kiwi.