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44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings. (2003)
Cambridge, Massachusettes
Oct. 11, 2003 to Oct. 14, 2003
ISSN: 0272-5428
ISBN: 0-7695-2040-5
pp: 116
Lászl Babai , University of Chicago
Amir Shpilka , Harvard University
Daniel Štefanković , University of Chicago
<p>Cyclic linear codes of block length n over a finite field \mathbb{F}_q are 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]).</p> <p>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 <li>(i) if x \in C then x is surely accepted;</li> <li>if dist(x,C) \geqslant \varepsilon n then x is probably rejected.</li> <li>("dist" refers to Hamming distane.)</li></p> <p>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.</p> <p>In this paper we address the intersection of the two questions stated.</p>

L. Babai, A. Shpilka and D. Štefanković, "Locally Testable Cyclic Codes," 44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings.(FOCS), Cambridge, Massachusettes, 2003, pp. 116.
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