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
Randomized Interpolation and Approximationof Sparse Polynomials
SIAM Journal on Computing
Pseudorandom generators without the XOR Lemma (extended abstract)
STOC '99 Proceedings of the thirty-first annual ACM symposium on Theory of computing
Learning Polynomials with Queries: The Highly Noisy Case
SIAM Journal on Discrete Mathematics
Near-optimal sparse fourier representations via sampling
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Proving Hard-Core Predicates Using List Decoding
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
Correcting Errors Beyond the Guruswami-Sudan Radius in Polynomial Time
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
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
IEEE Transactions on Information Theory
List decoding tensor products and interleaved codes
Proceedings of the forty-first annual ACM symposium on Theory of computing
An algorithm for computing a basis of a finite abelian group
CAI'11 Proceedings of the 4th international conference on Algebraic informatics
Linear time algorithms for the basis of abelian groups
COCOON'11 Proceedings of the 17th annual international conference on Computing and combinatorics
List Decoding Tensor Products and Interleaved Codes
SIAM Journal on Computing
Hi-index | 0.00 |
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 abelian 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 is the following combinatorial result: There is a fixed polynomial p(•) such that for every pair of abelian groups G and H, if the maximum fraction of agreement between two distinct homomorphisms from G to H is Λ, then for every ε 0 and every function f:G - H, the number of homomorphisms that have agreement Λ + ε with f is at most p(1/ε). We thus give a broad class of codes whose list-decoding radius exceeds the "Johnson bound". Examples of such codes are rare in the literature, and for the ones that do exist, "combinatorial" techniques to analyze their list-decodability are limited. Our work is an attempt to add to the body of such techniques. We use the fact that abelian groups decompose into simpler ones and thus codes derived from homomorphisms over abelian groups may be viewed as certain "compositions" of simpler codes. We give techniques to lift list-decoding bounds for the component codes to bounds for the composed code. We believe these techniques may be of general interest.