TY - GEN

T1 - Locally testable cyclic codes

AU - Babai, L.

AU - Shpilka, A.

AU - Štefankovič, D.

N1 - Publisher Copyright:
© 2003 IEEE.

PY - 2003

Y1 - 2003

N2 - Cyclic linear codes of block length n over a finite field double-struck Fq are the linear subspaces of double-struck Fqn 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 based on F.J. MacWilliams and N.J.A. Sloane (1977). A code C is r-testable if there exist a randomized algorithm which, given a word x ∈ double-struck Fqn, 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 ∈ C then x is surely accepted; (ii) if dist(x,C) ≥ εn then x is probably rejected ("dist" refers to Hamming distance). A family of codes is locally testable if all members of the family are r-testable for some constant r. This concept arose from holographic proofs/PCPs. O. Goldreich and M. Sudan (2002) asked whether there exist good, locally testable families of codes. In this paper we address the intersection of the two questions stated.

AB - Cyclic linear codes of block length n over a finite field double-struck Fq are the linear subspaces of double-struck Fqn 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 based on F.J. MacWilliams and N.J.A. Sloane (1977). A code C is r-testable if there exist a randomized algorithm which, given a word x ∈ double-struck Fqn, 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 ∈ C then x is surely accepted; (ii) if dist(x,C) ≥ εn then x is probably rejected ("dist" refers to Hamming distance). A family of codes is locally testable if all members of the family are r-testable for some constant r. This concept arose from holographic proofs/PCPs. O. Goldreich and M. Sudan (2002) asked whether there exist good, locally testable families of codes. In this paper we address the intersection of the two questions stated.

KW - Computer science

KW - Testing

UR - http://www.scopus.com/inward/record.url?scp=17744386581&partnerID=8YFLogxK

U2 - 10.1109/SFCS.2003.1238186

DO - 10.1109/SFCS.2003.1238186

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AN - SCOPUS:0344118897

T3 - Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS

SP - 116

EP - 125

BT - Proceedings - 44th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2003

PB - IEEE Computer Society

T2 - 44th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2003

Y2 - 11 October 2003 through 14 October 2003

ER -