Self-testing/correcting for polynomials and for approximate functions
STOC '91 Proceedings of the twenty-third annual ACM symposium on Theory of computing
The dimension of projective geometry codes
Discrete Mathematics - A collection of contributions in honour of Jack van Lint
Highly resilient correctors for polynomials
Information Processing Letters
STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
Improved low-degree testing and its applications
STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
Proof verification and the hardness of approximation problems
Journal of the ACM (JACM)
Robust Characterizations of Polynomials withApplications to Program Testing
SIAM Journal on Computing
Some improvements to total degree tests
ISTCS '95 Proceedings of the 3rd Israel Symposium on the Theory of Computing Systems (ISTCS'95)
Testing Polynomials over General Fields
SIAM Journal on Computing
An Improved Analysis of Linear Mergers
Computational Complexity
2-Transitivity Is Insufficient for Local Testability
CCC '08 Proceedings of the 2008 IEEE 23rd Annual Conference on Computational Complexity
Computational Complexity: A Modern Approach
Computational Complexity: A Modern Approach
Testing low-degree polynomials over prime fields
Random Structures & Algorithms
Succinct Representation of Codes with Applications to Testing
APPROX '09 / RANDOM '09 Proceedings of the 12th International Workshop and 13th International Workshop on Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques
Extensions to the Method of Multiplicities, with Applications to Kakeya Sets and Mergers
FOCS '09 Proceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science
Optimal Testing of Reed-Muller Codes
FOCS '10 Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
High-rate codes with sublinear-time decoding
Proceedings of the forty-third annual ACM symposium on Theory of computing
Symmetric LDPC Codes are not Necessarily Locally Testable
CCC '11 Proceedings of the 2011 IEEE 26th Annual Conference on Computational Complexity
Kakeya Sets, New Mergers, and Old Extractors
SIAM Journal on Computing
Optimal Testing of Multivariate Polynomials over Small Prime Fields
FOCS '11 Proceedings of the 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science
Edge transitive ramanujan graphs and symmetric LDPC good codes
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
IEEE Transactions on Information Theory
A new family of locally correctable codes based on degree-lifted algebraic geometry codes
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
Local correctability of expander codes
ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part I
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In this work we explore error-correcting codes derived from the "lifting" of "affine-invariant" codes. Affine-invariant codes are simply linear codes whose coordinates are a vector space over a field and which are invariant under affine-transformations of the coordinate space. Lifting takes codes defined over a vector space of small dimension and lifts them to higher dimensions by requiring their restriction to every subspace of the original dimension to be a codeword of the code being lifted. While the operation is of interest on its own, this work focusses on new ranges of parameters that can be obtained by such codes, in the context of local correction and testing. In particular we present four interesting ranges of parameters that can be achieved by such lifts, all of which are new in the context of affine-invariance and some may be new even in general. The main highlight is a construction of high-rate codes with sublinear time decoding. The only prior construction of such codes is due to Kopparty, Saraf and Yekhanin [33]. All our codes are extremely simple, being just lifts of various parity check codes (codes with one symbol of redundancy), and in the final case, the lift of a Reed-Solomon code. We also present a simple connection between certain lifted codes and lower bounds on the size of "Nikodym sets". Roughly, a Nikodym set in Fqm is a set S with the property that every point has a line passing through it which is almost entirely contained in S. While previous lower bounds on Nikodym sets were roughly growing as qm/2m, we use our lifted codes to prove a lower bound of (1 - o(1))qm for fields of constant characteristic.