TY - JOUR

T1 - Locally testable cyclic codes

AU - Babai, László

AU - Shpilka, Amir

AU - Štefankovič, Daniel

N1 - Funding Information:
Manuscript received December 2, 2003; revised February 15, 2005. The work of A. Shpilka was supported by the National Security Agency, Advanced Research and Development Activity under Army Research Office Contract DAAD19-01-1-0506 and by the Koshland Fellowship. The material in this paper was presented in part at the 44th IEEE Symposium on Foundations of Computer Science, Cambridge, MA, October 2003. Part of this work was done while A. Shpilka was a Postdoctoral Fellow at the Division of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138 USA and the Laboratory for Computer Science, MIT, Cambridge, MA 02139 USA.

PY - 2005/8

Y1 - 2005/8

N2 - Cyclic linear codes of block length n over a finite field Fq are linear subspaces of 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. A code C is r-testable if there exists a randomized algorithm which, given a word x ∈ 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) ≥ ∈n 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/PCP's. Recently it was asked whether there exist good, locally testable families of codes. In this paper the intersection of the two questions stated is addressed. Theorem. There are no good, locally testable families of cyclic codes over any (fixed) finite field. In fact the result is stronger in that it replaces condition ii) of local testability by the condition ii') if dist(x, C) ≥ ∈n then x has a positive chance of being rejected. The proof involves methods from Galois theory, cyclotomy, and diophantine approximation.

AB - Cyclic linear codes of block length n over a finite field Fq are linear subspaces of 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. A code C is r-testable if there exists a randomized algorithm which, given a word x ∈ 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) ≥ ∈n 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/PCP's. Recently it was asked whether there exist good, locally testable families of codes. In this paper the intersection of the two questions stated is addressed. Theorem. There are no good, locally testable families of cyclic codes over any (fixed) finite field. In fact the result is stronger in that it replaces condition ii) of local testability by the condition ii') if dist(x, C) ≥ ∈n then x has a positive chance of being rejected. The proof involves methods from Galois theory, cyclotomy, and diophantine approximation.

KW - Coding theory

KW - Cyclic codes

KW - Locally testable codes

UR - http://www.scopus.com/inward/record.url?scp=23844458042&partnerID=8YFLogxK

U2 - 10.1109/TIT.2005.851735

DO - 10.1109/TIT.2005.851735

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AN - SCOPUS:23844458042

SN - 0018-9448

VL - 51

SP - 2849

EP - 2858

JO - IEEE Transactions on Information Theory

JF - IEEE Transactions on Information Theory

IS - 8

ER -