Simultaneous diophantine approximation with excluded primes
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
Almost Orthogonal Linear Codes are Locally Testable
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
Locally testable vs. locally decodable codes
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
Tolerant locally testable codes
APPROX'05/RANDOM'05 Proceedings of the 8th international workshop on Approximation, Randomization and Combinatorial Optimization Problems, and Proceedings of the 9th international conference on Randamization and Computation: algorithms and techniques
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
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Cyclic linear codes of block length n over a finite field \mathbb{F}_qare the linear subspace of \mathbb{F}_{_q }^n that are invariant under a cyclic shift of their coordinates. A family of codes is good if all the codes in the family have constant rate and constant normalized distance (distance divided by block length). It is a long-standing open problem whether there exists a good family of cyclic linear codes (cf. [MS, p. 270]).A code C is r-testable if there exist a randomized algorithm which, given a word x \in \mathbb{F}_q^n, adaptively selects r positions, checks the entries of x in the selected positions, and makes a decision (accept or reject x) based on the positions selected and the numbers found, such that(i) if x \in C then x is surely accepted;if dist(x,C) \geqslant \varepsilon n then x is probably rejected.("dist" refers to Hamming distane.)A family of codes is locally testable if all members of the family are r-testable for some some constant r. This concept arose from holographic proofs/PCPs. Goldreich and Sudan [GS] asked whether there exist good, locally testable families of codes.In this paper we address the intersection of the two questions stated.